1.05 Data distribution

Data may be described as symmetrical or asymmetrical.

There are many cases where the data tends to be around a central value with no bias left or right. In such a case, roughly 50 \% of scores will be above the mean and 50 \% of scores will be below the mean. In other words, the mean and median roughly coincide.

The normal distribution is a common example of a symmetrical distribution of data. The normal distribution looks like this bell-shaped curve.

This picture shows how a data set that has an approximate normal distribution may appear in a histogram. The dark line shows the nice, symmetrical curve that can be drawn over the histogram that the data roughly follows.

In the distribution below, the 0 point in the middle represents the mean, the median and the mode - all of these measures of central tendency are equal for this distribution, since it is symmetrical. If there was a line at 0 as our axis of symmetry, notice that the left-hand side is a perfect reflection of the right-hand side.

If a data set is asymmetrical instead (i.e. it isn't symmetrical), it may be described as skewed.

A data set that has positive skew (sometimes called a 'right skew') has a longer tail of values above the peak of the graph, such that more than half of the scores are above the peak. This means the mean is greater than the median, which is greater than the mode: \text{mode} < \text{median} < \text{mean}

A positively skewed graph looks something like this:

Notice that there are more scores above the peak than below the peak.

A data set that has negative skew (sometimes called a 'left skew') has a longer tail of values below the peak of the graph, such that more than half of the scores are below the peak. This means the mode is greater than the median, which is greater than the mean: \text{mean} < \text{median} < \text{mode}

A negatively skewed graph looks something like this:

Notice that there are more scores below the peak than above the peak.

Examples

Example 1

State whether the scores in each histogram are positively skewed, negatively skewed or symmetrical (approximately).

a

A

Positively skewed

B

Negatively skewed

C

Symmetrical

Worked Solution

Create a strategy

Check where the bulk of the data sits and look at the general shape of the distribution.

Apply the idea

The scores are roughly even in both the high and low end, so the distribution is symmetrical, option C.

Watch question walkthrough

b

A

Positively skewed

B

Negatively skewed

C

Symmetrical

Worked Solution

Create a strategy

Check where the bulk of the data sits and look at the general shape of the distribution.

Apply the idea

Most of the scores on the histogram are relatively low, so the distribution is positively skewed, option A.

Watch question walkthrough

c

A

Symmetrical

B

Negatively skewed

C

Positively skewed

Worked Solution

Create a strategy

Check where the bulk of the data sits and look at the general shape of the distribution.

Apply the idea

Most of the scores on the histogram are relatively high, so the distribution is negatively skewed, option B.

Watch question walkthrough

In a set of data, a cluster occurs when a large number of the scores are grouped together within a very small range.

The shape of data also shows us whether there are any outliers or unusually high or low values in a data set.

For example, in the dot plot below, do you see how all the ages range between 12 and 14 except one? This means that 24 is an outlier.

In this case, the outlier is very obviously way outside the range of the rest of the data set.

The formal definition of an outlier is a score that is more than 1.5 \times \text{IQR} above the upper quartile, or less than 1.5 \times \text{IQR} below the lower quartile. This will be discussed further in a later section.

Modality describes the prevalence of local peaks in a data set. The peaks don't necessarily need to be the mode of the whole data set, but rather a local cluster of data that is more frequent and stands out from the surrounding data. When looking at the modality of a data set, it is usually useful to examine a graph of the data. Modality is described by the number of peaks.

A data set that has two distinct peaks, like in the histogram below, is called bimodal.

To determine the modality of a distribution, simply identify the number of modal peaks. For instance, the data shown in the dot plot below has three modal peaks because there are local peaks at scores of 6,\,12, and 20.

Examples

Example 2

For the Stem and Leaf plot attached:

Stem	Leaf
0	5
1	7\ 8
2	0\ 8
3	1\ 3\ 3\ 7\ 8\ 9
4	1\ 3\ 5\ 8\ 8\ 8
5
6
7
8
9	2
Key 1\vert 2 = 12

a

Are there any outliers?

Worked Solution

Create a strategy

Look for significantly large or small values in the data set.

Apply the idea

Looking at the stem, we can see that after 4, we have 9, which is significantly larger. So, yes, there is an outlier.

Reflect and check

Another way to look for an outlier is to find the values of upper and lower quartile. The upper quartile is 45, and the lower quartile is 28. We take the difference to get an \text{IQR} of 17. We can now see if any numbers in the data set are more than 45 + 1.5 \times 17 = 70.5, or less than 28- 1.5 \times 17 = 2.5.

Watch question walkthrough

b

Identify the outlier.

Worked Solution

Create a strategy

Use the answer in part (a).

Apply the idea

In part (a), we found an outlier and looking again at the stem and leaf plot, 92 is the outlier. \text{Outlier}=92

Reflect and check

To be an outlier the number has to be more than 1.5 \times \text{IQR} above the upper quartile or below the lower quartile. In this case, any outlier must be more than 70.5 or less than 2.5.

Watch question walkthrough

c

Is there any clustering of data?

Worked Solution

Create a strategy

Look for a large number of the scores grouped within a small range.

Apply the idea

Looking at the stem, both 3 and 4 have 6 leaf values which can be identified as clustering - where the large portion of the scores lie. So, yes, there is clustering in the data set.

Watch question walkthrough

d

Where does the clustering occur?

A

10s to 20s

B

30s to 40s

C

20s to 30s

Worked Solution

Create a strategy

Use the answers in part (c)

Apply the idea

In part (c), we found that stems 3 and 4 have clustering of data. So, the correct answer is B.

Watch question walkthrough

e

What is the modal class(es)?

A

10-19

B

40-49

C

30-39

D

20-29

Worked Solution

Create a strategy

Look at the class that occurs most often. Note that there may be more than one modal class.

Apply the idea

We found in part (c) that stems 3 and 4 have the same number of data points, and both have the highest frequency.

So the correct answers are options B and C.

Watch question walkthrough

f

The distribution of the data is:

A

Positively skewed

B

Symmetrical

C

Negatively skewed

Worked Solution

Create a strategy

Ignore the outlier and check if the where the most of the scores lie.

Apply the idea

Most of the scores are relatively high in the data set, which makes a negatively skewed distribution. So the correct answer is C.

Watch question walkthrough

Example 3

How many peaks are there on the graph?

Worked Solution

Create a strategy

Count how many high points are in the data distribution.

Apply the idea

Scores on 6,\,12, and 20 have high points or frequency in the data distribution, which makes them the peaks. So we have 3 peaks on the graph.

Watch question walkthrough

It is important to note that certain features in a data set can significantly affect one or more of the three measures of central tendency (the mean, median and mode).

Remember the mode is the most frequently occurring score. So, if a data set has a significant number of repeated scores, then the mode could be a good measure of centre.

If the range of scores is reasonably small and there are no outliers, then the mean is an appropriate measure of centre.

Unlike the mean, the median is not affected by outliers. So, the median is a good measure of central tendency if a data set has outliers or a large range.

The shape of the data may also determine which measure of central tendency is the most appropriate measure of a data set.

If a data set is symmetrical, then the mean and median will be approximately equal. If the data is unimodal (has only one mode) then the mode will also be approximately equal. If the data has more than one mode (e.g. if it is bimodal) then the modes may be different to the mean and median.

When data is positively skewed, the mean is the highest measure of central tendency and the mode is the lowest measure of central tendency. For positively skewed data: \text{mode}<\text{median}<\text{mean}

When data is negatively skewed, the mode is the highest measure of central tendency and the mean is the lowest measure of central tendency. For negatively skewed data: \text{mean}<\text{median}<\text{mode}

Therefore, in skewed data, the most appropriate measure of central tendency will be the median.

Here is a basic summary of selecting an appropriate measure of central tendency. Note that often it can be helpful to consider more than one measure.

Of course, sometimes the context of the data being analysed lends itself to particular measures as well.

Examples