7.04 Measures of spread

Measures of spread in a numerical data set seek to describe whether the scores in a data set are very similar and clustered together, or whether there is a lot of variation in the scores and they are very spread out.

There are several methods to describe the spread of data, which vary greatly in complexity. It is possible to simply look at the numerical range of the entire data set, or break the data into chunks. The spread of data can also be compared to the mean, which can then be normalised for a meaningful comparison to other data sets.

 

Range

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$7 vary between $11$11 and $14$14. So the range for this set is $14-11=3$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$7 year old and the oldest person might be a $90$90 year old. The range of this set of data is $90-7=83$90−7=83, which is a much larger range of ages.

Remember, the range only changes if the highest or lowest score in a data set is changed. Otherwise, it will remain the same.

 

Practice questions

Question 1

Question 2

Question 3

 

Interquartile range

Whilst the range is very simple to calculate, it is based on the sparse information provided by the upper and lower limits of the data set. To get a better picture of the internal spread in a data set, it is often more useful to find the set's quartiles, from which the interquartile range (IQR) can be calculated.

Quartiles are scores at particular locations in the data set–similar to the median, but instead of dividing a data set into halves, they divide a data set into quarters. Let's look at how to divide up some data sets into quarters now.

Exploration

• Here is a data set with $8$8 scores:

$\editable{1}$1

$\editable{3}$3

$\editable{4}$4

$\editable{7}$7

$\editable{11}$11

$\editable{12}$12

$\editable{14}$14

$\editable{19}$19

 

First locate the median, between the $4$4th and $5$5th scores:

Median

$\downarrow$↓

$\editable{1}$1

$\editable{3}$3

$\editable{4}$4

$\editable{7}$7

$\editable{11}$11

$\editable{12}$12

$\editable{14}$14

$\editable{19}$19

 

Now there are $4$4 scores in each half of the data set, so split each of the four scores in half to find the quartiles. We can see the first quartile ($Q_1$Q1​) is between the $2$2nd and $3$3rd scores; there are two scores on either side of $Q_1$Q1​. Similarly, the upper quartile ($Q_3$Q3​) is between the $6$6th and $7$7th scores:

$Q_1$Q1​

Median

$Q_3$Q3​

$\downarrow$↓

$\downarrow$↓

$\downarrow$↓

$\editable{1}$1

$\editable{3}$3

$\editable{4}$4

$\editable{7}$7

$\editable{11}$11

$\editable{12}$12

$\editable{14}$14

$\editable{19}$19

 

 

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

$Q_1$Q1​

Median

$Q_3$Q3​

$\downarrow$↓

$\downarrow$↓

$\downarrow$↓

$\editable{8}$8

$\editable{8}$8

$\editable{10}$10

$\editable{11}$11

$\editable{13}$13

$\editable{14}$14

$\editable{18}$18

$\editable{22}$22

$\editable{25}$25

This time, the $5$5th term is the median. There are four terms on either side of the median, like for the set with eight scores. So $Q_1$Q1​ is still between the $2$2nd and $3$3rd scores and $Q_3$Q3​ is between the $6$6th and $7$7th scores.

 

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

$Q_1$Q1​

Median

$Q_3$Q3​

$\downarrow$↓

$\downarrow$↓

$\downarrow$↓

$\editable{12}$12

$\editable{13}$13

$\editable{14}$14

$\editable{19}$19

$\editable{19}$19

$\editable{21}$21

$\editable{22}$22

$\editable{22}$22

$\editable{28}$28

$\editable{30}$30

For this set, the median is between the $5$5th and $6$6th scores. This time, however, there are $5$5 scores on either side of the median. So $Q_1$Q1​ is the $3$3rd term and $Q_3$Q3​ is the $8$8th term.

 

What do the quartiles represent?

Each quartile represents $25%$25% of the data set. The lowest score to the first quartile represents $25%$25% of the data; the first quartile to the median represents another $25%$25%; the median to the third quartile is another $25%$25%; and the third quartile to the highest score represents the last $25%$25% of the data. It is possible to combine these quartiles together - for example, $50%$50% of the scores in a data set lie between the first and third quartiles.

These quartiles are sometimes named as percentiles. A percentile is a percentage that indicates the value below which a given percentage of observations in a group of observations fall. For example, if a score is in the $75$75th percentile in a statistical test, it is higher than $75%$75% of all other scores. The median represents the $50$50th percentile, or the halfway point in a data set.

 

Naming the quartiles

 

• $Q_1$Q1​ is the first quartile (sometimes called the lower quartile). It is the middle score in the bottom half of data and it represents the $25$25th percentile.
• $Q_2$Q2​ is the second quartile, and is usually called the median, which we have already learnt about. It represents the $50$50th percentile of the data set.
• $Q_3$Q3​ is the third quartile (sometimes called the upper quartile). It is the middle score in the top half of the data set, and represents the $75$75th percentile.

 

Calculating the interquartile range

The interquartile range (IQR) is the difference between the third quartile and the first quartile. $50%$50% of scores lie within the IQR because two full quartiles lie in this range. Since it focuses on the middle $50%$50% of the data set, the interquartile range often gives a better indication of the internal spread than the range does, and it is less affected by individual scores that are unusually high or low (called outliers).

Worked example

Example 1

Consider the following set of data: $1,1,3,5,7,9,9,10,15$1,1,3,5,7,9,9,10,15.

(a) Identify the median.

Think: There are nine numbers in the set, so we can say that $n=9$n=9. We can also see that the data set is already arranged in ascending order. We identify the median as the middle score either by the "cross-out" method or as the $\frac{n+1}{2}$n+12​th score. 

Do:

$\text{Position of median}$Position of median	$=$=	$\frac{9+1}{2}$9+12​	Substituting $n=9$n=9 into $\frac{n+1}{2}$n+12​
	$=$=	$5$5th score	Simplifying the fraction

 

Counting through the set to the $5$5th score gives us $7$7 as the median.

(b) Identify $Q_1$Q1​ (the lower quartile) and $Q_3$Q3​ (the upper quartile).

Think: We identify $Q_1$Q1​ and $Q_3$Q3​ as the middle scores in the lower and upper halves of the data set respectively, either by the "cross-out" method–or any method that we use to find the median, but just applying it to the lower or upper half of the data set. 

Do: The lower half of the data set is all the scores to the left of the median, which is $1,1,3,5$1,1,3,5. There are four scores here, so $n=4$n=4. So we can find the position of $Q_1$Q1​ as follows:

$\text{Position of }Q_1$Position of Q1​	$=$=	$\frac{4+1}{2}$4+12​	Substituting $n=4$n=4 into $\frac{n+1}{2}$n+12​
	$=$=	$2.5$2.5th score	Simplifying the fraction

 

$Q_1$Q1​ is therefore the mean of the $2$2nd and $3$3rd scores. So we see that:

$Q_1$Q1​	$=$=	$\frac{1+3}{2}$1+32​	Taking the average of the $2$2nd and $3$3rd scores
	$=$=	$2$2	Simplifying the fraction

 

The upper half of the data set is all the scores to the right of the median, which is $9,9,10,15$9,9,10,15. Since there are also $n=4$n=4 scores, $Q_3$Q3​ will be the mean of the $2$2nd and $3$3rd scores in this upper half.

$Q_3$Q3​	$=$=	$\frac{9+10}{2}$9+102​	Taking the average of the $2$2nd and $3$3rd scores in the upper half
	$=$=	$9.5$9.5	Simplifying the fraction

 

(c) Calculate the $\text{IQR }$IQR  of the data set.

Think: Remember that $\text{IQR }=Q_3-Q_1$IQR =Q3​−Q1​, and we just found $Q_1$Q1​ and $Q_3$Q3​.

Do:

$\text{IQR }$IQR	$=$=	$9.5-2$9.5−2	Substituting $Q_1=9.5$Q1​=9.5 and $Q_3=2$Q3​=2 into the formula
	$=$=	$7.5$7.5	Simplifying the subtraction

 

 

Practice questions

Question 4

Question 5

Question 6