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Line 1,791: − === How do you construct a histogram from a continuous variable? ===+ + + + + + + + + + + + − To construct a+ − − into intervals, called bins. In the example above, age has been split − into bins, with each bin representing a 10-year period starting at 20 − years. Each bin contains the number of occurrences of scores in the − data set that are contained within that bin. For the above data set, − the frequencies in each bin have been tabulated along with the scores − that contributed to the frequency in each bin (see below): − − + − Bin Frequency Scores+ − Included in Bin+ + + + + − 20-30 2 25,22+ − 30-40 4 36,38,36,38+ − 40-50 4 46,45,48,46+ − 50-60 5 55,55,52,58,55+ − − 60-70 3 68,67,61+ + + + + − 70-80 1 72 − − 80-90 0 - − 90-100 1 91+ + + + + + + + + − Notice that, unlike a+ − bar chart, there are no "gaps" between the bars (although − some bars might be "absent" reflecting no frequencies). − This is because a histogram represents a continuous data set, and as − such, there are no gaps in the data. (Although you will have to − decide whether you round up or round down scores on the boundaries of − bins) − − + + + + − + − + − There is no right or+ − wrong answer as to how wide a bin should be, but there are rules of − thumb. You need to make sure that the bins are not too small or too − large. Consider the histogram we produced earlier (see above): the − following histograms use the same data but have either much smaller − or larger bins, as shown below: − − + − − [[Image:KOER-%20Mathematics%20-%20Statistics_html_75ab55c3.png]]+ − − − We can see from the+ − histogram on the left, that the bin width is too small as it shows+ − too much individual data and does not allow the underlying pattern+ − (frequency distribution) of the data to be easily seen. At the other+ − end of the scale, is the diagram on the right, where the bins are too+ − large and, again, we are unable to find the underlying trend in the − data. − + − Histograms are based on+ − area not height of bars+ + + − + + − In a histogram, it is+ − the area of the bar that indicates the frequency of occurrences for+ − each bin. This means that the height of the bar does not necessarily+ − indicate how many occurrences of scores there were within each − individual bin. It is the product of height multiplied by the width − of the bin that indicates the frequency of occurrences within that − bin. One of the reasons that the height of the bars is often − incorrectly assessed as indicating frequency and not the area of the − bar is due to the fact that a lot of histograms often have equally − spaced bars (bins) and, under these circumstances, the height of the − bin does reflect the frequency. − + + − === What is the difference between a bar chart and a histogram? ===+ + + + − [[Image:KOER-%20Mathematics%20-%20Statistics_html_6dfca87b.png]]+ + − The major difference is+ − that a histogram is only used to plot the frequency of score+ − occurrences in a continuous data set that has been divided into+ − classes, called bins. Bar charts, on the other hand, can be used for − a great deal of other types of variables including ordinal and − nominal data sets. + + + + + + + + + + + + + + + + + + − == Circle or Pie Chart ==+ + + − [[Image:KOER-%20Mathematics%20-%20Statistics_html_461389d1.png]]These+ − are called circle graphs. A circle graph shows the relationship+ − between a whole and its parts. Here, the whole circle is divided into+ − sectors. The size of each sector is proportional to the activity or − information it represents. + + + + + + − A variety of graphical+ − representations of data are now possible using spreadsheet software.+ − OpenOffice CALC can convert a table of data into bar charts, pie − charts, area charts etc and make data much more easy to − read/interpret. + + + + + + − − == Activities == − === Activity 2: Histogram and Bar Chart ===+ + + + − ==== Learning Objectives ====+ − + − Learn to draw a histogram and bar chart. − Understand the difference between a bar chart and a histogram and be − able to select the appropriate chart by looking at the problem and − data. − ==== Materials and Resources Required ==== − − Paper and Pencil − − − ==== Pre-requisites/ Instructions ==== − − ==== Method ==== − − Solve the problems A and B − − − − − − − A> In the past year, you have recorded the − number of tickets that a movie theater has sold during each month. − To represent this data set graphically, would you construct a bar − graph or a histogram? Why is this choice better than the other? − Using the following data, construct the graph that you choose. − − − {| border="1" − Month+ − Number of Tickets Sold+ − January+ − 25+ − + − | + − February+ + + − | + − 20+ + + + + + + + + + + − March+ − 15+ − April+ − 20+ − May+ − 30+ − June+ − 35+ − July+ − 40+ − August+ − 20+ − September+ − 25+ − October+ − 15+ − November+ − 20+ − December+ − 30+ − + − + + − + + − B> For a recent+ − science project, you collected data regarding the distribution of+ − fish and aquatic life in a nearby pond. Your data consists of the+ − number of living creatures found in each 1 meter depth increment in+ − the pond. Construct a bar graph and several histograms (vary the+ − depth increment size) for the following data. In which case(s) is the+ − histogram the same as the bar graph? How do the other histograms vary+ − from the bar graph?+ + + + + + + + + + + + + + + + + + − − {| border="1" − |- − | − '''Depth Range''' − | + − '''Number of Living Creatures '''+ − + + + − |-+ − | + − 0 – 1 meters − | + − 10+ + + + + + + + − |-+ − | + − 1 – 2 meters − | + − 93+ − |- − | − 2 – 3 meters − + − | − 23 − |-+ − | + − 3 – 4 meters − | − 47 − − |- − | − 4 – 5 meters − − | − 68 − |-+ − | + − 5 – 6 meters+ + + + + + + + + − | − 51 − − |- − | − 6 – 7 meters − − | − 43 − |-+ − | + − 7 – 8 meters+ + + + + − | + − 21+ − |-+ − | + − 8 – 9 meters+ − | + − 15+ − − |- − | − 9 – 10 meters − − | − 8 − |} + − ==== Evaluation ====+ − + − # Does the student understand the difference between a bar chart and a histogram ?+ − # Does the student know when to use each of these charts - - depending on the type of data continuous and discrete ?+ − + − == Evaluation ==+ − + − == Self-Evaluation ==+ − + − == Further Explorations ==+ − + − === Types of Variables ===+ − + − All experiments examine some kind of variable(s).+ − A variable is not only something that we measure, but also something+ − that we can manipulate and something we can control for. To − understand the characteristics of variables and how we use them in − research, this guide is divided into three main sections. First, we − illustrate the role of dependent and independent variables. Second, − we discuss the difference between experimental and non-experimental − research. Finally, we explain how variables can be characterised as − either categorical or continuous. − === Dependent and Independent Variables ===+ + + + + + + + + + + + + + + + + + + Line 2,241:
Line 2,279: − An independent variable, sometimes called an+ − experimental or predictor variable, is a variable that is being − manipulated in an experiment in order to observe the effect on a − dependent variable, sometimes called an outcome variable. − Line 2,251:
Line 2,285: − Imagine that a tutor asks 100 students to complete+ − a maths test. The tutor wants to know why some students perform+ − better than others. Whilst the tutor does not know the answer to+ − this, she thinks that it might be because of two reasons: (1) some − students spend more time revising for their test; and (2) some − students are naturally more intelligent than others. As such, the − tutor decides to investigate the effect of revision time and − intelligence on the test performance of the 100 students. The − dependent and independent variables for the study are: Line 2,266:
Line 2,294: − Dependent Variable: Test Mark (measured from 0 to+ − 100)+ + + + + + + + + + − + − + + + + + + + + + − Independent Variables: Revision time (measured in+ − hours) Intelligence (measured using IQ score)+ + + + + + + + + + + + Line 2,282:
Line 2,338: − The dependent variable is simply that, a variable+ − that is dependent on an independent variable(s). For example, in our+ − case the test mark that a student achieves is dependent on revision+ − time and intelligence. Whilst revision time and intelligence (the − independent variables) may (or may not) cause a change in the test − mark (the dependent variable), the reverse is implausible; in other − words, whilst the number of hours a student spends revising and the − higher a student's IQ score may (or may not) change the test mark − that a student achieves, a change in a student's test mark has no − bearing on whether a student revises more or is more intelligent − (this simply doesn't make sense). Line 2,299:
Line 2,347: − Therefore, the aim of the tutor's investigation is+ − to examine whether these independent variables - revision time and IQ+ − - result in a change in the dependent variable, the students' test+ − scores. However, it is also worth noting that whilst this is the main+ − aim of the experiment, the tutor may also be interested to know if+ − the independent variables - revision time and IQ - are also connected+ − − + − + − + − + − In the section on experimental and+ − non-experimental research that follows, we find out a little more+ − about the nature of independent and dependent variables.+ + + + + − − === Experimental and Non-Experimental Research === Line 2,323:
Line 2,372: − Experimental research: In experimental research,+ − the aim is to manipulate an independent variable(s) and then examine+ − the effect that this change has on a dependent variable(s). Since it+ − is possible to manipulate the independent variable(s), experimental+ − research has the advantage of enabling a researcher to identify a − cause and effect between variables. For example, take our example of − 100 students completing a maths exam where the dependent variable was − the exam mark (measured from 0 to 100) and the independent variables − were revision time (measured in hours) and intelligence (measured − using IQ score). Here, it would be possible to use an experimental − design and manipulate the revision time of the students. The tutor − could divide the students into two groups, each made up of 50 − students. In "group one", the tutor could ask the students − not to do any revision. Alternately, "group two" could be − asked to do 20 hours of revision in the two weeks prior to the test. − The tutor could then compare the marks that the students achieved. − Non-experimental research: In non-experimental+ − research, the researcher does not manipulate the independent+ − + − it will either be impractical or unethical to do so. For example, a+ − researcher may be interested in the effect of illegal, recreational+ − drug use (the dependent variable(s)) on certain types of behaviour+ − (the independent variable(s)). However, whilst possible, it would be+ − unethical to ask individuals to take illegal drugs in order to study+ − what effect this had on certain behaviours. As such, a researcher+ − could ask both drug and non-drug users to complete a questionnaire − that had been constructed to indicate the extent to which they − exhibited certain behaviours. Whilst it is not possible to identify − the cause and effect between the variables, we can still examine the − association or relationship between them.In addition to understanding − the difference between dependent and independent variables, and − experimental and non-experimental research, it is also important to − understand the different characteristics amongst variables. This is − discussed next. Line 2,364:
Line 2,392: − − === Categorical and Continuous Variables === Line 2,371:
Line 2,397: − Categorical variables are also known as discrete+ − or qualitative variables. Categorical variables can be further+ − categorized as either''' nominal, ordinal or dichotomous.'''+ − − + − − − '''Nominal variables''' are variables that have+ − two or more categories but which do not have an intrinsic order. For+ − example, a real estate agent could classify their types of property+ − into distinct categories such as houses, condos, co-ops or bungalows.+ − So "type of property" is a nominal variable with 4+ − categories called houses, condos, co-ops and bungalows. Of note, the+ − different categories of a nominal variable can also be referred to as+ − groups or levels of the nominal variable. Another example of a+ − nominal variable would be classifying where people live in Karnataka − by district. In this case there will be many more levels of the − nominal variable (30 in fact). − '''Dichotomous variables''' are nominal+ − variables which have only two categories or levels. For example, if+ − we were looking at gender, we would most probably categorize somebody+ − as either "male" or "female". This is an example+ − + − example might be if we asked a person if they owned a mobile phone.+ − Here, we may categorise mobile phone ownership as either "Yes" − or "No". In the real estate agent example, if type of − property had been classified as either residential or commercial then − "type of property" would be a dichotomous variable. − '''Ordinal variables''' are variables that have+ − two or more categories just like nominal variables only the+ − categories can also be ordered or ranked. So if you asked someone if+ − they liked the policies of the Democratic Party and they could answer+ − either "Not very much", "They are OK" or "Yes,+ − a lot" then you have an ordinal variable. Why? Because you have+ − 3 categories, namely "Not very much", "They are OK"+ − − positive (Yes, a lot), to the middle response (They are OK), to the − least positive (Not very much). However, whilst we can rank the − levels, we cannot place a "value" to them; we cannot say − that "They are OK" is twice as positive as "Not very − much" for example. − + + + + + + + + + − + + − Continuous variables are also known as+ − quantitative variables. Continuous variables can be further+ − categorized as either interval or ratio variables.+ − + − − '''Interval variables''' are variables for which+ − their central characteristic is that they can be measured along a+ − continuum and they have a numerical value (for example, temperature+ − measured in degrees Celsius or Fahrenheit). So the difference between+ − 20C and 30C is the same as 30C to 40C. However, temperature measured+ − in degrees Celsius or Fahrenheit is NOT a ratio variable.+ + + − '''Ratio variables''' are interval variables but+ − with the added condition that 0 (zero) of the measurement indicates − that there is none of that variable. So, temperature measured in − degrees Celsius or Fahrenheit is not a ratio variable because 0C does − not mean there is no temperature. However, temperature measured in − Kelvin is a ratio variable as 0 Kelvin (often called absolute zero) − indicates that there is no temperature whatsoever. Other examples of − ratio variables include height, mass, distance and many more. The − name "ratio" reflects the fact that you can use the ratio − of measurements. So, for example, a distance of ten metres is twice − the distance of 5 metres. − + − + + + + + + − === Ambiguities in classifying a type of variable ===+ + + + + Line 2,464:
Line 2,483: − In some cases, the measurement scale for data is+ − ordinal but the variable is treated as continuous. For example, a − Likert scale that contains five values - strongly agree, agree, − neither agree nor disagree, disagree, and strongly disagree - is − ordinal. However, where a Likert scale contains seven or more value - − strongly agree, moderately agree, agree, neither agree nor disagree, − disagree, moderately disagree, and strongly disagree - the underlying − scale is sometimes treated as continuous although where you should do − this is a cause of great dispute. + + + + + + + + + + + + + + + + + + + + + − == Enrichment Activities ==+ + + − = Central tendency =+ + + − == Introduction ==+ + + − A measure of central tendency is a single value+ − that attempts to describe a set of data by identifying the central+ − position within that set of data. As such, measures of central − tendency are sometimes called measures of central location. They are − also classed as summary statistics. The mean (often called the − average) is most likely the measure of central tendency that you are − most familiar with, but there are others, such as, the median and the − mode. − The mean, median and mode are all valid measures+ − of central tendency but, under different conditions, some measures of+ − central tendency become more appropriate to use than others. In the − following sections we will look at the mean, mode and median and − learn how to calculate them and under what conditions they are most − appropriate to be used. − == Objectives ==+ + + − * Understand and know that a measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data.+ − * Understand that the mean, median and mode are all valid measures of central tendency but, under different conditions, some measures of central tendency become more appropriate to use than others.+ − * Learn to calculation of mean and median and analyse data and make conclusions.+ + − == Mean (Arithmetic) ==+ + + − The mean (or average) is the most popular and well+ − known measure of central tendency. It can be used with both discrete+ − and continuous data, although its use is most often with continuous − data. The mean is equal to the sum of all the values in the data set − divided by the number of values in the data set. So, if we have n − values in a data set and they have values x<sub>1</sub>, x<sub>2</sub>, − ..., x<sub>n</sub>, then the sample mean, usually denoted by − [[Image:KOER-%20Mathematics%20-%20Statistics_html_174cec39.gif]] − (pronounced x bar), is: + + + + + + + + − [[Image:KOER-%20Mathematics%20-%20Statistics_html_69b2cf9e.gif]]+ + − This formula is usually written in a slightly+ − different manner using the Greek capitol letter, Σ,+ − pronounced "sigma", which means "sum of...": − [[Image:KOER-%20Mathematics%20-%20Statistics_html_m50e9a786.gif]]+ + − You may have noticed that the above formula refers+ − to the sample mean. So, why call have we called it a sample mean?+ − This is because, in statistics, samples and populations have very − different meanings and these differences are very important, even if, − in the case of the mean, they are calculated in the same way. To − acknowledge that we are calculating the population mean and not the − sample mean, we use the Greek lower case letter "mu", − denoted as µ: − [[Image:KOER-%20Mathematics%20-%20Statistics_html_7b1e9596.gif]]+ + − + − It is the value that is most common. You will notice, however, that+ − the mean is not often one of the actual values that you have observed+ − in your data set. However, one of its important properties is that it+ − minimises error in the prediction of any one value in your data set.+ − That is, it is the value that produces the lowest amount of error+ − from all other values in the data set.+ + + − An important property of the mean is that it+ − includes every value in your data set as part of the calculation. In+ − addition, the mean is the only measure of central tendency where the+ − sum of the deviations of each value from the mean is always zero.+ − + + + + + + + + + + + + + + + + − − '''When not to use the mean''' − + − The mean has one main disadvantage: it is − particularly susceptible to the influence of outliers. These are − values that are unusual compared to the rest of the data set by being − especially small or large in numerical value. For example, consider − the wages of staff at a factory below: − − − Staff+ − 1+ − 2+ − 3+ − 4+ − 5+ − 6+ − 7+ − 8+ − 9+ − 10+ + + + + + + + + + + + + + + − Salary+ − 15k+ − 18k+ − 16k+ − 14k+ − 15k+ − 15k+ − 12k+ − 17k+ − 90k+ − 95k+ − The mean salary for these ten staff is $30.7k. − However, inspecting the raw data suggests that this mean value might − not be the best way to accurately reflect the typical salary of a − worker, as most workers have salaries in the $12k to 18k range. The − mean is being skewed by the two large salaries. Therefore, in this − situation we would like to have a better measure of central tendency. − As we will find out later, taking the median would be a better − measure of central tendency in this situation. − + − Another time when we usually prefer the median − over the mean (or mode) is when our data is skewed (i.e. the − frequency distribution for our data is skewed). If we consider the − normal distribution - as this is the most frequently assessed in − statistics - when the data is perfectly normal then the mean, median − and mode are identical. Moreover, they all represent the most typical − value in the data set. However, as the data becomes skewed the mean − loses its ability to provide the best central location for the data − as the skewed data is dragging it away from the typical value. − However, the median best retains this position and is not as strongly − influenced by the skewed values. This is explained in more detail in − the skewed distribution section later in this guide. − == Median ==+ − + − The median is the middle score for a set of data+ − that has been arranged in order of magnitude. The median is less+ − affected by outliers and skewed data. In order to calculate the+ − median, suppose we have the data below:+ + Line 2,710:
Line 2,743: − + Line 2,751:
Line 2,784: − − − | − 92 Line 2,762:
Line 2,791: − + − − Line 2,789:
Line 2,816: − + Line 2,821:
Line 2,848: − Our median mark is the middle mark - in this case+ − 56 (highlighted in bold). It is the middle mark because there are 5+ − scores before it and 5 scores after it. This works fine when you have − an odd number of scores but what happens when you have an even number − of scores? What if you had only 10 scores? Well, you simply have to − − the example below: + + + + + + − + − + − − {| border="1" − |- − | − 65 − | + − 55+ + − | + − 89+ + + + − + − | + − 56 − + − | + − 35+ − + − + − | + − 14+ + + + + − + − | + − 56+ + − | + − 55 − | + − 87+ + + + − | + − 45 − − |} + + + + − + − We again rearrange that data into order of+ − magnitude (smallest first):+ − + − + − + − + − {| border="1"+ − |- − | − 14 − | + − 35+ − | + − 45 − − − | − 55 − − − | − '''55''' − − − | − '''56''' − − − | − 56 − − − | − 65 − − − | − 87 − − − | − 89 − − − | − 92 − − − |} − − − − − Only now we have to take the 5th and 6th score in − our data set and average them to get a median of 55.5. − − − == Mode == − − The mode is the most frequent score in our data − set. On a histogram it represents the highest bar in a bar chart or − histogram. You can, therefore, sometimes consider the mode as being − the most popular option. An example of a mode is presented below: − − − [[Image:KOER-%20Mathematics%20-%20Statistics_html_58d59706.png]] − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − Normally, the mode is used for categorical data − where we wish to know which is the most common category as − illustrated below: − − − We can see above that the most common form of − transport, in this particular data set, is the bus. However, one of − the problems with the mode is that it is not unique, so it leaves us − with problems when we have two or more values that share the highest − frequency, such as below: − − − − − − − [[Image:KOER-%20Mathematics%20-%20Statistics_html_m64bbad46.png]] − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − We are now stuck as to which mode best describes − the central tendency of the data. This is particularly problematic − when we have continuous data, as we are more likely not to have any − one value that is more frequent than the other. For example, consider − measuring 30 peoples' weight (to the nearest 0.1 kg). How likely is − it that we will find two or more people with '''exactly''' − the same weight, e.g. 67.4 kg? The answer, is probably very unlikely − - many people might be close but with such a small sample (30 people) − and a large range of possible weights you are unlikely to find two − people with exactly the same weight, that is, to the nearest 0.1 kg. − This is why the mode is very rarely used with continuous data. − − − − − − − Another problem with the mode is that it will not − provide us with a very good measure of central tendency when the most − common mark is far away from the rest of the data in the data set, as − depicted in the diagram below: − − − [[Image:KOER-%20Mathematics%20-%20Statistics_html_152dd141.png]] − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − In the above diagram the mode has a value of 2. We − can clearly see, however, that the mode is not representative of the − data, which is mostly concentrated around the 20 to 30 value range. − To use the mode to describe the central tendency of this data set − would be misleading. − − − == Skewed Distributions and the Mean and Median == − − [[Image:KOER-%20Mathematics%20-%20Statistics_html_26c6186d.png]]We − often test whether our data is normally distributed as this is a − common assumption underlying many statistical tests. An example of a − normally distributed set of data is presented below: − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − When you have a normally distributed sample you − can legitimately use both the mean or the median as your measure of − central tendency. In fact, in any symmetrical distribution the mean, − median and mode are equal. However, in this situation, the mean is − widely preferred as the best measure of central tendency as it is the − measure that includes all the values in the data set for its − calculation, and any change in any of the scores will affect the − value of the mean. This is not the case with the median or mode. − − − However, when our data is skewed, for example, as − with the right-skewed data set below: − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −
no edit summary
= Introduction =
= Introduction =
=== Descriptive and Inferential Statistics ===
=== Descriptive and Inferential Statistics ===
When analysing data, for example, the marks
When analysing data, for example, the marks
[[Image:KOER-%20Mathematics%20-%20Statistics_html_6201ec25.png]]
36 25 38 46 55 68
72 55 36 38
67 45 22 48 91 46
52 61 58 55
=== How do you construct a histogram from a continuous variable? ===
To construct a
histogram from a continuous variable you first need to split the data
into intervals, called bins. In the example above, age has been split
into bins, with each bin representing a 10-year period starting at 20
years. Each bin contains the number of occurrences of scores in the
data set that are contained within that bin. For the above data set,
the frequencies in each bin have been tabulated along with the scores
that contributed to the frequency in each bin (see below):
Bin Frequency Scores
Included in Bin
20-30 2 25,22
30-40 4 36,38,36,38
40-50 4 46,45,48,46
50-60 5 55,55,52,58,55
60-70 3 68,67,61
70-80 1 72
80-90 0 -
90-100 1 91
Notice that, unlike a
bar chart, there are no "gaps" between the bars (although
some bars might be "absent" reflecting no frequencies).
This is because a histogram represents a continuous data set, and as
such, there are no gaps in the data. (Although you will have to
decide whether you round up or round down scores on the boundaries of
bins)
=== Choosing the correct bin width ===
There is no right or
wrong answer as to how wide a bin should be, but there are rules of
thumb. You need to make sure that the bins are not too small or too
large. Consider the histogram we produced earlier (see above): the
following histograms use the same data but have either much smaller
or larger bins, as shown below:
[[Image:KOER-%20Mathematics%20-%20Statistics_html_75ab55c3.png]]
We can see from the
histogram on the left, that the bin width is too small as it shows
too much individual data and does not allow the underlying pattern
(frequency distribution) of the data to be easily seen. At the other
end of the scale, is the diagram on the right, where the bins are too
large and, again, we are unable to find the underlying trend in the
data.
Histograms are based on
area not height of bars
In a histogram, it is
the area of the bar that indicates the frequency of occurrences for
each bin. This means that the height of the bar does not necessarily
indicate how many occurrences of scores there were within each
individual bin. It is the product of height multiplied by the width
of the bin that indicates the frequency of occurrences within that
bin. One of the reasons that the height of the bars is often
incorrectly assessed as indicating frequency and not the area of the
bar is due to the fact that a lot of histograms often have equally
spaced bars (bins) and, under these circumstances, the height of the
bin does reflect the frequency.
=== What is the difference between a bar chart and a histogram? ===
histogram from a continuous variable you first need to split the data
[[Image:KOER-%20Mathematics%20-%20Statistics_html_6dfca87b.png]]
The major difference is
that a histogram is only used to plot the frequency of score
occurrences in a continuous data set that has been divided into
classes, called bins. Bar charts, on the other hand, can be used for
a great deal of other types of variables including ordinal and
nominal data sets.
== Circle or Pie Chart ==
[[Image:KOER-%20Mathematics%20-%20Statistics_html_461389d1.png]]These
are called circle graphs. A circle graph shows the relationship
between a whole and its parts. Here, the whole circle is divided into
sectors. The size of each sector is proportional to the activity or
information it represents.
A variety of graphical
representations of data are now possible using spreadsheet software.
OpenOffice CALC can convert a table of data into bar charts, pie
charts, area charts etc and make data much more easy to
read/interpret.
== Activities ==
=== Activity 2: Histogram and Bar Chart ===
==== Learning Objectives ====
Learn to draw a histogram and bar chart.
Understand the difference between a bar chart and a histogram and be
able to select the appropriate chart by looking at the problem and
data.
=== Choosing the correct bin width ===
==== Materials and Resources Required ====
Paper and Pencil
==== Pre-requisites/ Instructions ====
==== Method ====
Solve the problems A and B
A> In the past year, you have recorded the
number of tickets that a movie theater has sold during each month.
To represent this data set graphically, would you construct a bar
graph or a histogram? Why is this choice better than the other?
Using the following data, construct the graph that you choose.
{| border="1"
|-
|
Month
|
Number of Tickets Sold
|-
|
January
|
25
|-
|
February
|
20
|-
|
March
|
15
|-
|
April
|
20
|-
|
May
|
30
|-
|
June
|
35
|-
|
July
|
40
|-
|
August
|
20
|-
|
September
|
25
|-
|
October
|
15
|-
|-
|
|
November
|
|
20
|-
|-
|
|
December
|
|
30
|-
|}
B> For a recent
science project, you collected data regarding the distribution of
fish and aquatic life in a nearby pond. Your data consists of the
number of living creatures found in each 1 meter depth increment in
the pond. Construct a bar graph and several histograms (vary the
depth increment size) for the following data. In which case(s) is the
histogram the same as the bar graph? How do the other histograms vary
from the bar graph?
{| border="1"
|-
|-
|
|
'''Depth Range'''
|
|
'''Number of Living Creatures '''
|-
|-
|
|
0 – 1 meters
|
|
10
|-
|-
|
|
1 – 2 meters
|
|
93
|-
|-
|
|
2 – 3 meters
|
|
23
|-
|-
|
|
3 – 4 meters
|
|
47
|-
|-
|
|
4 – 5 meters
|
|
68
|-
|-
|
|
5 – 6 meters
|
|
51
|-
|-
|
|
6 – 7 meters
|
|
43
|-
|-
|
|
7 – 8 meters
|
|
21
|-
|-
|
|
8 – 9 meters
|
|
15
|}
|-
|
9 – 10 meters
|
8
|}
==== Evaluation ====
# Does the student understand the difference between a bar chart and a histogram ?
# Does the student know when to use each of these charts - - depending on the type of data continuous and discrete ?
== Evaluation ==
== Self-Evaluation ==
== Further Explorations ==
=== Types of Variables ===
All experiments examine some kind of variable(s).
A variable is not only something that we measure, but also something
that we can manipulate and something we can control for. To
understand the characteristics of variables and how we use them in
research, this guide is divided into three main sections. First, we
illustrate the role of dependent and independent variables. Second,
we discuss the difference between experimental and non-experimental
research. Finally, we explain how variables can be characterised as
either categorical or continuous.
=== Dependent and Independent Variables ===
An independent variable, sometimes called an
experimental or predictor variable, is a variable that is being
manipulated in an experiment in order to observe the effect on a
dependent variable, sometimes called an outcome variable.
Imagine that a tutor asks 100 students to complete
a maths test. The tutor wants to know why some students perform
better than others. Whilst the tutor does not know the answer to
this, she thinks that it might be because of two reasons: (1) some
students spend more time revising for their test; and (2) some
students are naturally more intelligent than others. As such, the
tutor decides to investigate the effect of revision time and
intelligence on the test performance of the 100 students. The
dependent and independent variables for the study are:
Dependent Variable: Test Mark (measured from 0 to
100)
Independent Variables: Revision time (measured in
hours) Intelligence (measured using IQ score)
The dependent variable is simply that, a variable
that is dependent on an independent variable(s). For example, in our
case the test mark that a student achieves is dependent on revision
time and intelligence. Whilst revision time and intelligence (the
independent variables) may (or may not) cause a change in the test
mark (the dependent variable), the reverse is implausible; in other
words, whilst the number of hours a student spends revising and the
higher a student's IQ score may (or may not) change the test mark
that a student achieves, a change in a student's test mark has no
bearing on whether a student revises more or is more intelligent
(this simply doesn't make sense).
Therefore, the aim of the tutor's investigation is
to examine whether these independent variables - revision time and IQ
- result in a change in the dependent variable, the students' test
scores. However, it is also worth noting that whilst this is the main
aim of the experiment, the tutor may also be interested to know if
the independent variables - revision time and IQ - are also connected
in some way.
In the section on experimental and
non-experimental research that follows, we find out a little more
about the nature of independent and dependent variables.
=== Experimental and Non-Experimental Research ===
Experimental research: In experimental research,
the aim is to manipulate an independent variable(s) and then examine
the effect that this change has on a dependent variable(s). Since it
is possible to manipulate the independent variable(s), experimental
research has the advantage of enabling a researcher to identify a
cause and effect between variables. For example, take our example of
100 students completing a maths exam where the dependent variable was
the exam mark (measured from 0 to 100) and the independent variables
were revision time (measured in hours) and intelligence (measured
using IQ score). Here, it would be possible to use an experimental
design and manipulate the revision time of the students. The tutor
could divide the students into two groups, each made up of 50
students. In "group one", the tutor could ask the students
not to do any revision. Alternately, "group two" could be
asked to do 20 hours of revision in the two weeks prior to the test.
The tutor could then compare the marks that the students achieved.
Non-experimental research: In non-experimental
research, the researcher does not manipulate the independent
variable(s). This is not to say that it is impossible to do so, but
it will either be impractical or unethical to do so. For example, a
researcher may be interested in the effect of illegal, recreational
drug use (the dependent variable(s)) on certain types of behaviour
(the independent variable(s)). However, whilst possible, it would be
unethical to ask individuals to take illegal drugs in order to study
what effect this had on certain behaviours. As such, a researcher
could ask both drug and non-drug users to complete a questionnaire
that had been constructed to indicate the extent to which they
exhibited certain behaviours. Whilst it is not possible to identify
the cause and effect between the variables, we can still examine the
association or relationship between them.In addition to understanding
the difference between dependent and independent variables, and
experimental and non-experimental research, it is also important to
understand the different characteristics amongst variables. This is
discussed next.
=== Categorical and Continuous Variables ===
Categorical variables are also known as discrete
or qualitative variables. Categorical variables can be further
categorized as either''' nominal, ordinal or dichotomous.'''
'''Nominal variables''' are variables that have
two or more categories but which do not have an intrinsic order. For
example, a real estate agent could classify their types of property
into distinct categories such as houses, condos, co-ops or bungalows.
So "type of property" is a nominal variable with 4
categories called houses, condos, co-ops and bungalows. Of note, the
different categories of a nominal variable can also be referred to as
groups or levels of the nominal variable. Another example of a
nominal variable would be classifying where people live in Karnataka
by district. In this case there will be many more levels of the
nominal variable (30 in fact).
'''Dichotomous variables''' are nominal
variables which have only two categories or levels. For example, if
we were looking at gender, we would most probably categorize somebody
as either "male" or "female". This is an example
of a dichotomous variable (and also a nominal variable). Another
example might be if we asked a person if they owned a mobile phone.
Here, we may categorise mobile phone ownership as either "Yes"
or "No". In the real estate agent example, if type of
property had been classified as either residential or commercial then
"type of property" would be a dichotomous variable.
'''Ordinal variables''' are variables that have
two or more categories just like nominal variables only the
categories can also be ordered or ranked. So if you asked someone if
they liked the policies of the Democratic Party and they could answer
either "Not very much", "They are OK" or "Yes,
a lot" then you have an ordinal variable. Why? Because you have
3 categories, namely "Not very much", "They are OK"
and "Yes, a lot" and you can rank them from the most
positive (Yes, a lot), to the middle response (They are OK), to the
least positive (Not very much). However, whilst we can rank the
levels, we cannot place a "value" to them; we cannot say
that "They are OK" is twice as positive as "Not very
much" for example.
Continuous variables are also known as
quantitative variables. Continuous variables can be further
categorized as either interval or ratio variables.
'''Interval variables''' are variables for which
their central characteristic is that they can be measured along a
continuum and they have a numerical value (for example, temperature
measured in degrees Celsius or Fahrenheit). So the difference between
20C and 30C is the same as 30C to 40C. However, temperature measured
in degrees Celsius or Fahrenheit is NOT a ratio variable.
in some way.
'''Ratio variables''' are interval variables but
with the added condition that 0 (zero) of the measurement indicates
that there is none of that variable. So, temperature measured in
degrees Celsius or Fahrenheit is not a ratio variable because 0C does
not mean there is no temperature. However, temperature measured in
Kelvin is a ratio variable as 0 Kelvin (often called absolute zero)
indicates that there is no temperature whatsoever. Other examples of
ratio variables include height, mass, distance and many more. The
name "ratio" reflects the fact that you can use the ratio
of measurements. So, for example, a distance of ten metres is twice
the distance of 5 metres.
=== Ambiguities in classifying a type of variable ===
In some cases, the measurement scale for data is
ordinal but the variable is treated as continuous. For example, a
variable(s). This is not to say that it is impossible to do so, but
Likert scale that contains five values - strongly agree, agree,
neither agree nor disagree, disagree, and strongly disagree - is
ordinal. However, where a Likert scale contains seven or more value -
strongly agree, moderately agree, agree, neither agree nor disagree,
disagree, moderately disagree, and strongly disagree - the underlying
scale is sometimes treated as continuous although where you should do
this is a cause of great dispute.
== Enrichment Activities ==
= Central tendency =
== Introduction ==
A measure of central tendency is a single value
that attempts to describe a set of data by identifying the central
position within that set of data. As such, measures of central
tendency are sometimes called measures of central location. They are
also classed as summary statistics. The mean (often called the
average) is most likely the measure of central tendency that you are
most familiar with, but there are others, such as, the median and the
mode.
The mean, median and mode are all valid measures
of central tendency but, under different conditions, some measures of
central tendency become more appropriate to use than others. In the
following sections we will look at the mean, mode and median and
of a dichotomous variable (and also a nominal variable). Another
learn how to calculate them and under what conditions they are most
appropriate to be used.
== Objectives ==
* Understand and know that a measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data.
* Understand that the mean, median and mode are all valid measures of central tendency but, under different conditions, some measures of central tendency become more appropriate to use than others.
* Learn to calculation of mean and median and analyse data and make conclusions.
== Mean (Arithmetic) ==
and "Yes, a lot" and you can rank them from the most
The mean (or average) is the most popular and well
known measure of central tendency. It can be used with both discrete
and continuous data, although its use is most often with continuous
data. The mean is equal to the sum of all the values in the data set
divided by the number of values in the data set. So, if we have n
values in a data set and they have values x<sub>1</sub>, x<sub>2</sub>,
..., x<sub>n</sub>, then the sample mean, usually denoted by
[[Image:KOER-%20Mathematics%20-%20Statistics_html_174cec39.gif]]
(pronounced x bar), is:
[[Image:KOER-%20Mathematics%20-%20Statistics_html_69b2cf9e.gif]]
This formula is usually written in a slightly
different manner using the Greek capitol letter, Σ,
pronounced "sigma", which means "sum of...":
[[Image:KOER-%20Mathematics%20-%20Statistics_html_m50e9a786.gif]]
You may have noticed that the above formula refers
to the sample mean. So, why call have we called it a sample mean?
This is because, in statistics, samples and populations have very
different meanings and these differences are very important, even if,
in the case of the mean, they are calculated in the same way. To
acknowledge that we are calculating the population mean and not the
sample mean, we use the Greek lower case letter "mu",
denoted as µ:
[[Image:KOER-%20Mathematics%20-%20Statistics_html_7b1e9596.gif]]
The mean is essentially a model of your data set.
It is the value that is most common. You will notice, however, that
the mean is not often one of the actual values that you have observed
in your data set. However, one of its important properties is that it
minimises error in the prediction of any one value in your data set.
That is, it is the value that produces the lowest amount of error
from all other values in the data set.
An important property of the mean is that it
includes every value in your data set as part of the calculation. In
addition, the mean is the only measure of central tendency where the
sum of the deviations of each value from the mean is always zero.
'''When not to use the mean'''
The mean has one main disadvantage: it is
particularly susceptible to the influence of outliers. These are
values that are unusual compared to the rest of the data set by being
especially small or large in numerical value. For example, consider
the wages of staff at a factory below:
{| border="1"
|-
|
Staff
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|-
|
Salary
|
15k
|
18k
|
16k
|
14k
|
15k
|
15k
|
12k
|
17k
|
90k
|
95k
The mean is essentially a model of your data set.
|}
The mean salary for these ten staff is $30.7k.
However, inspecting the raw data suggests that this mean value might
not be the best way to accurately reflect the typical salary of a
worker, as most workers have salaries in the $12k to 18k range. The
mean is being skewed by the two large salaries. Therefore, in this
situation we would like to have a better measure of central tendency.
As we will find out later, taking the median would be a better
measure of central tendency in this situation.
Another time when we usually prefer the median
over the mean (or mode) is when our data is skewed (i.e. the
frequency distribution for our data is skewed). If we consider the
normal distribution - as this is the most frequently assessed in
statistics - when the data is perfectly normal then the mean, median
and mode are identical. Moreover, they all represent the most typical
value in the data set. However, as the data becomes skewed the mean
loses its ability to provide the best central location for the data
as the skewed data is dragging it away from the typical value.
However, the median best retains this position and is not as strongly
influenced by the skewed values. This is explained in more detail in
the skewed distribution section later in this guide.
== Median ==
The median is the middle score for a set of data
that has been arranged in order of magnitude. The median is less
affected by outliers and skewed data. In order to calculate the
median, suppose we have the data below:
{| border="1"
{| border="1"
|-
|-
|
|
65
|
|
55
|
|
89
|
|
56
|
|
35
|
|
14
|
|
56
|
|
55
|
|
87
|
|
45
|
|
92
|}
We first need to rearrange that data into order of
magnitude (smallest first):
{| border="1"
|-
|-
|
|
14
|
|
35
|
|
45
|
|
55
|
|
55
|
|
'''56'''
|
|
56
|
|
65
|
|
87
|
|
89
|
|
92
|}
|}
Our median mark is the middle mark - in this case
56 (highlighted in bold). It is the middle mark because there are 5
scores before it and 5 scores after it. This works fine when you have
an odd number of scores but what happens when you have an even number
of scores? What if you had only 10 scores? Well, you simply have to
take the middle two scores and average the result. So, if we look at
the example below:
{| border="1"
{| border="1"
|-
|-
|
|
45
45
We first need to rearrange that data into order of
We again rearrange that data into order of
magnitude (smallest first):
magnitude (smallest first):
|
|
55
'''55'''
Only now we have to take the 5th and 6th score in
our data set and average them to get a median of 55.5.
take the middle two scores and average the result. So, if we look at
== Mode ==
The mode is the most frequent score in our data
set. On a histogram it represents the highest bar in a bar chart or
histogram. You can, therefore, sometimes consider the mode as being
the most popular option. An example of a mode is presented below:
[[Image:KOER-%20Mathematics%20-%20Statistics_html_58d59706.png]]
Normally, the mode is used for categorical data
where we wish to know which is the most common category as
illustrated below:
We can see above that the most common form of
transport, in this particular data set, is the bus. However, one of
the problems with the mode is that it is not unique, so it leaves us
with problems when we have two or more values that share the highest
frequency, such as below:
[[Image:KOER-%20Mathematics%20-%20Statistics_html_m64bbad46.png]]
We are now stuck as to which mode best describes
the central tendency of the data. This is particularly problematic
when we have continuous data, as we are more likely not to have any
one value that is more frequent than the other. For example, consider
measuring 30 peoples' weight (to the nearest 0.1 kg). How likely is
it that we will find two or more people with '''exactly'''
the same weight, e.g. 67.4 kg? The answer, is probably very unlikely
- many people might be close but with such a small sample (30 people)
and a large range of possible weights you are unlikely to find two
people with exactly the same weight, that is, to the nearest 0.1 kg.
This is why the mode is very rarely used with continuous data.
Another problem with the mode is that it will not
provide us with a very good measure of central tendency when the most
common mark is far away from the rest of the data in the data set, as
depicted in the diagram below:
[[Image:KOER-%20Mathematics%20-%20Statistics_html_152dd141.png]]
In the above diagram the mode has a value of 2. We
can clearly see, however, that the mode is not representative of the
data, which is mostly concentrated around the 20 to 30 value range.
To use the mode to describe the central tendency of this data set
would be misleading.
== Skewed Distributions and the Mean and Median ==
[[Image:KOER-%20Mathematics%20-%20Statistics_html_26c6186d.png]]We
often test whether our data is normally distributed as this is a
common assumption underlying many statistical tests. An example of a
normally distributed set of data is presented below:
When you have a normally distributed sample you
can legitimately use both the mean or the median as your measure of
central tendency. In fact, in any symmetrical distribution the mean,
median and mode are equal. However, in this situation, the mean is
widely preferred as the best measure of central tendency as it is the
measure that includes all the values in the data set for its
calculation, and any change in any of the scores will affect the
value of the mean. This is not the case with the median or mode.
However, when our data is skewed, for example, as
with the right-skewed data set below:
[[Image:KOER-%20Mathematics%20-%20Statistics_html_m2609c500.png]]
[[Image:KOER-%20Mathematics%20-%20Statistics_html_m2609c500.png]]