1.03 Measures of centre and spread

The mean is often referred to as the average. To calculate the mean, add all the scores in a data set, then divide this by number of scores.

The mean of a data set, denoted \overline{x}, is given by \overline{x}=\dfrac{\Sigma x}{n}.

The median is the middle score in a data set, when the data is sorted to be in order. If there are two middle scores, we take the average of them to find the median.

To find the median of a data set:

• Sort the data points into order

• If there are an odd number of scores, the median is the middle score (the \dfrac{n+1}{2} score

• If there are an even number of scores, the median is calculated by finding the mean of the two middle scores (the \dfrac{n}{2}th and \left(\dfrac{n}{2} + 1\right) th scores.)

The mode describes the most frequently occurring score.

Suppose that 10 people were asked how many pets they had. 2 people said they didn't own any pets, 6 people had one pet and 2 people said they had two pets.

In this data set, the most common number of pets that people have is one pet, and so the mode of this data set is 1.

A data set can have more than one mode, if two or more scores are equally tied as the most frequently occurring.

The mode of a data set is the most frequently occurring score.

Examples

The range is the simplest measure of spread in a quantitative (numerical) data set. It is the difference between the maximum and minimum scores in a data set.

For example, at one school the ages of students in Year 7 vary between 11 and 14. So the range for this set is 14-11=3.

As a different example, if we looked at the ages of people waiting at a bus stop, the youngest person might be a 7 year old and the oldest person might be a 90 year old. The range of this set of data is 90-7=83, which is a much larger range of ages.

The Range of a numerical data set is given by: \text{Range }=\text{maximum score}-\text{minimum score}.

When we are trying to the understand what our data is telling us, we might want to find measures of centre (the median, mean, and modes) as well as measures of spread, such as the range. However, the range is easily affected by outliers. To get a better picture of the spread in a data set, we can instead find the set's quartiles.

Each quartile represents 25\% of the data set. The Lower quartile marks 25\% of the way through data. The median, which we already know, marks 50\% of the way through the data, and the upper quartile marks 75\% of the way through the data. We can use these to measure the spread of the data. For example, 50\% of the scores in a data set lie between the lower and upper quartiles.

We already know how to calculate the median of a data set. Calculating quartiles is exactly the same, but we use only the lower half of the scores (for the lower quartile), or the upper half of the scores (for the upper quartiles).

Here is a data set with 8 scores:

First locate the median, between the 4\text{th} and 5\text{th} scores:

Now there are four scores in each half of the data set, so split each of the four scores in half to find the quartiles. We can see the first quartile, Q_{1} is between the 2\text{nd} and 3\text{rd} scores, so there are two scores on either side of Q_{1}. Similarly, the third quartile, Q_{3} is between the 6\text{th} and 7\text{th} scores:

Now let's look at a situation with 9 scores:

This time, the 5\text{th} term is the median. There are four terms on either side of the median, like for the set with eight scores. So Q_{1} is still between the 2\text{nd} and 3\text{rd} scores and Q_{3} is between the 6\text{th} and 7\text{th} scores.

Finally, let's look at a set with 10 scores:

For this set, the median is between the 5\text{th} and 6\text{th} scores. This time, however, there are 5 scores on either side of the median. So Q_{1} is the 3\text{rd} term and Q_{3} is the 8\text{th} term.

• The first quartile is also called the lower quartile. It is the middle score between the lowest score and the median and it represents the 25th percentile. The first quartile score is the \dfrac{n+1}{4}th score, where n is the total number of scores.

• The second quartile is the median, which we have already learnt about and it represents the 50th percentile.

• The third quartile is also called the upper quartile. It is the middle score between the median and the highest score. It represents the 75th percentile. The third quartile is the \dfrac{3(n+1)}{4}th score, where n is the total number of scores.

The interquartile range (IQR) is the difference between the upper quartile and the lower quartile. 50\% of scores lie within the IQR because two full quartiles lie in this range.

Subtract the first quartile from the third quartile. That is, IQR = Q_{3}-Q_{1}

Examples