1.06 Measures of spread

Measures of spread in a quantitative (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.

This section will define the range, interquartile range, and standard deviation as measures of spread. How to break data into quartiles of any number is also explored.

 

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

The range of a set of scores is $8$8, and the highest score is $19$19.

What is the lowest score in the set?

Reveal SolutionWatch video

 

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

The first quartile ($Q_1$Q1​)

$Q_1$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.

 

The median

$Q_2$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.

 

The third quartile ($q_3$q3​)

$Q_3$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 could 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 eliminating scores at both ends until we reach the middle or as the $\frac{n+1}{2}$n+12​th score.

Do:

$\text{Middle position}$Middle position	$=$=	$\frac{9+1}{2}$9+12​
	$=$=	$5$5th score

Counting through the set to the $5$5th score, this means that the median is $7$7.

(b) Identify $Q_1$Q1​ (lower quartile) and $Q_3$Q3​ (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. We know that the lower and upper half contains $4$4 scores each. So we can find the middle of each set, as we would by finding the median.

Do:

$Q_1$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​
	$=$=	$2$2

Similarly,

$Q_3$Q3​ is therefore the mean of the $7$7th and $8$8th scores. So we see that

$Q_3$Q3​	$=$=	$\frac{9+10}{2}$9+102​
	$=$=	$9.5$9.5

(c) Calculate the $IQR$IQR of the data set.

Think: We remember that $IQR=Q_3-Q_1$IQR=Q3​−Q1​, and we just found $Q_1$Q1​ and $Q_3$Q3​.

Do:

$IQR$IQR	$=$=	$9.5-2$9.5−2
	$=$=	$7.5$7.5

Practice questions

Question 2

Question 3

 

Standard deviation

Standard deviation is a measure of spread, which helps give a meaningful estimate of the variability in a data set. While the quartiles were related to the median central tendency, the standard deviation is instead related to the mean central tendency. A small standard deviation indicates that most scores are close to the mean, while a large standard deviation indicates that the scores are more spread out away from the mean value.

The standard deviation can be calculated for a population or a sample.

The symbols used are:

$\text{Population Standard Deviation}$Population Standard Deviation	$=$=	$\sigma$σ	(lowercase sigma)
$\text{Sample Standard Deviation}$Sample Standard Deviation	$=$=	$s$s

In statistics mode on a calculator, the following symbols might be used:

$\text{Population Standard Deviation}$Population Standard Deviation	$=$=	$\sigma_n$σn​
$\text{Sample Standard Deviation}$Sample Standard Deviation	$=$=	$\sigma_{n-1}$σn−1​

Note: It is only required to calculate standard deviation using the automatic function in the statistics mode of our calculators, so we will not go through the formal definition and equation here.

The standard deviation is found by calculating the square root of the variance.

Variance is the average of the squared differences from the mean. Here is its formula.

$\sigma^2=\frac{1}{n}\Sigma\left(x_i-\mu\right)^2$σ2=1n​Σ(xi​−μ)2

This is the formula by which a calculator calculates the standard deviation of a data set from a full population. That is, it is the formula used for census data rather than sample data.

$\sigma=\sqrt{\frac{1}{n}\Sigma\left(x_i-\mu\right)^2}$σ=√1n​Σ(xi​−μ)2

In this formula, the numbers $x_i$xi​ are the values in the data set. There is one value for each subscript $i$i.
There are $n$n numbers $x_i$xi​ in the data set. So, $i$i goes from $1$1 to $n$n in the summation.
The symbol $\mu$μ (Greek letter 'mu') is the population mean.
The Greek letter $\sigma$σ (sigma) is used for the population standard deviation.
The symbol $\Sigma$Σ (upper case sigma) is the summation symbol. 

Steps:

1. Calculate the mean.  $\mu=\frac{1}{n}\Sigma_{i=1}^n\ x_i$μ=1n​Σni=1​ xi​ 
2. Find the difference from the mean for each score. $x_i-\mu$xi​−μ
3. Square each of the differences.    $\left(x_i-\mu\right)^2$(xi​−μ)2
4. Sum the squared differences.  $\Sigma\left(x_i-\mu\right)^2$Σ(xi​−μ)2
5. Divide the sum by the number of scores. $\frac{1}{n}\Sigma\left(x_i-\mu\right)^2$1n​Σ(xi​−μ)2
6. Take the square root.  $\sigma=\sqrt{\frac{1}{n}\Sigma\left(x_i-\mu\right)^2}$σ=√1n​Σ(xi​−μ)2

 

Simply put, standard deviation describes the spread of data by comparing the distance of each score to the mean. It is complicated to calculate, but it gives a lot of information about the spread of data because it takes into account every data point in the set.

Standard deviation is also a very powerful way of comparing different data sets, particularly if there are different means and population numbers.

Practice questions

Question 4

Question 5