Measures of Spread

What spread tells us

An average describes a typical value, but it does not tell us how much the values vary. A measure of spread, also called a measure of dispersion, describes how widely scattered the data values are.

When comparing datasets measuring the same quantity in the same units, a smaller spread indicates greater consistency. Different measures focus on different features: the range uses the two endpoints, the interquartile range uses the middle half, and standard deviation uses every value.

Range: the distance between the endpoints

The range is the largest value minus the smallest value:

$$\text{range}=\text{largest value}-\text{smallest value}$$

For a set of seed-germination times, the shortest time was 5 days and the longest was 23 days. The range was therefore $23-5=18$ days. This is a difference, not the interval ‘5 to 23 days’.

Range is quick to calculate, but it depends entirely on the two extreme values. It does not reveal how the remaining observations are distributed between them.

Quartiles: finding dividing values in ordered data

Quartiles divide ordered data into four approximately equal parts. The lower quartile, $Q_1$, marks about 25% of the data; the median, , marks about 50%; and the , , marks about 75%. These percentages describe dividing positions, so repeated values can prevent an exact percentage from lying strictly below a quartile.

First put the data in numerical order, retaining every repeated value. One way to find the quartiles is to locate the median, leave it out of both halves when the number of values is odd, and then find the median of each half. The median of the lower half is $Q_1$; the median of the upper half is $Q_3$.

For the quartile calculations specified here, the number of observations, $n$, is one less than a multiple of four, such as 7, 11 or 15. The quartile positions are therefore whole numbers:

$$Q_1\text{ position}=\frac{n+1}{4},\qquad Q_2\text{ position}=\frac{n+1}{2},\qquad Q_3\text{ position}=\frac{3(n+1)}{4}$$

For 11 observations, these positions are $12/4=3$, $12/2=6$ and $3\times12/4=9$. Read the third, sixth and ninth values in the ordered list. The answers are the data values at those positions, not the numbers 3, 6 and 9 themselves.

The two methods agree here: excluding the sixth value leaves five values in each half. Their middle values are the third and ninth values of the original list. There is no need to interpolate between observations for these specified dataset sizes.

Interquartile range: the spread of the middle half

The interquartile range, or IQR, is the difference between the upper and :

$$\mathrm{IQR}=Q_3-Q_1$$

In the seed-germination data, the lower quartile was 7 days and the upper quartile was 9 days. The IQR was $9-7=2$ days: the middle half of the germination times occupied a much narrower interval than the full dataset.

The 23-day germination time made the range large, while all the other times were between 5 and 11 days. The IQR better represents the spread of the central observations because it does not depend directly on the largest and smallest values. Moving an extreme endpoint further away, without changing the quartiles, changes the range but not the IQR.

Percentiles: choosing a percentage position

Percentiles divide ordered data into 100 approximately equal parts. The 65th percentile marks the value at or below which approximately 65% of the observations lie. Percentiles are data values, not percentages themselves.

Quartiles are particular percentiles:

$$Q_1=P_{25},\qquad Q_2=P_{50},\qquad Q_3=P_{75}$$

For grouped data, percentiles can be estimated from a cumulative frequency graph. Cumulative frequency is a running total: it counts observations up to each boundary. To find $P_p$ for $N$ observations, calculate the target cumulative frequency $pN/100$. Find that number on the vertical axis, move horizontally to the graph, then vertically down to the data-value axis.

The following quiz-time data show how this works:

Data for Estimating a percentile from cumulative frequency

There are 48 students, so the target cumulative frequency for the 65th percentile is $0.65\times48=31.2$. The cumulative frequency rises from 30 to 41 in the interval $12<t\leq14$, so the percentile lies in that interval. Reading the connecting line gives an estimate of about 12.2 minutes.

This is an estimate because the individual times within each interval are unknown. A straight line between class boundaries treats the observations as evenly distributed within that class. The same reading method transfers to other cumulative frequency graphs: calculate the target frequency first, then read the corresponding data value.

Interpercentile and interdecile ranges — Higher tier

An interpercentile range is the difference between two specified percentiles. For example, the 30th-to-80th interpercentile range is . It measures the width of the interval containing approximately 50% of the observations between those dividing values. The 25th-to-75th interpercentile range is the IQR.

For pepper-harvest masses, the 2.5th percentile was 274.1 g and the 97.5th percentile was 532.4 g. The 2.5th-to-97.5th interpercentile range was $532.4-274.1=258.3$ g, describing the spread of the central 95% of the data.

Deciles divide ordered data into ten approximately equal parts. The first decile, $D_1$, is , the second is , and so on. An is the difference between two specified deciles.

For the same harvest data, $P_{10}=319.7$ g and $P_{90}=488.3$ g. The first-to-ninth interdecile range was therefore g. This describes the spread of the central 80%. Other decile pairs can be used, so identify the pair before subtracting.

Standard deviation: spread around the mean — Higher tier

Standard deviation, usually written $\sigma$, measures spread around the mean using every observation. A small standard deviation indicates that values are generally close to the mean; a larger one indicates greater spread. Extreme values can affect it because every value enters the calculation.

A deviation is the difference $x-\bar{x}$ between an observation and the mean. Simply adding these deviations would give zero: positive and negative deviations cancel. Squaring them prevents that cancellation. Standard deviation takes the square root of the average squared deviation:

$$\sigma=\sqrt{\frac{\sum(x-\bar{x})^2}{n}}$$

Here, $x$ represents an observation, $\bar{x}$ is the mean, $n$ is the number of observations, and $\sum$ means ‘add all of these terms’.

For the values 6, 9, 2, 11 and 5, the mean is $33/5=6.6$:

The squared deviations total 49.2, so $\sigma=\sqrt{49.2/5}\approx3.14$.

An equivalent formula is often quicker, especially when totals are supplied:

$$\sigma=\sqrt{\frac{\sum x^2}{n}-\left(\frac{\sum x}{n}\right)^2}$$

For the same values, $\sum x=33$ and $\sum x^2=36+81+4+121+25=267$. Substitution gives , the same result.

Standard deviation from a frequency table — Higher tier

Frequency tells us how many times a value occurs. Rather than writing out all the observations, multiply each contribution by its frequency, $f$. The number of observations is $\sum f$, not the number of rows:

$$\bar{x}=\frac{\sum fx}{\sum f}$$ $$\sigma=\sqrt{\frac{\sum f(x-\bar{x})^2}{\sum f}} =\sqrt{\frac{\sum fx^2}{\sum f}-\left(\frac{\sum fx}{\sum f}\right)^2}$$

For a survey of rabbit owners, the calculation can be organised as follows:

The mean is $60/20=3$ rabbits, and the standard deviation is $\sqrt{208/20-(60/20)^2}\approx1.18$ rabbits. This describes spread around the mean; it is not the average number of rabbits owned.

Range, quartiles, percentile values, IQR, , and standard deviation all retain the units of the original data. For standard deviation, the final square root restores those units after the deviations have been squared.