4.3 Measures of central tendency and dispersion

Measures of central tendency

A measure of central tendency is a single value that represents the centre of a data set. The three in common use are the mean, the median and the mode, and each describes "typical" in a different way.

The mean

The mean is the familiar "average": add all the values and divide by how many there are. For $n$ values $x_1, x_2, \ldots, x_n$, the mean is written $\bar{x}$ (read "x bar"):

When the data are the whole population, the mean is often written $\mu$ instead. The two symbols are calculated in the same way.

The mean uses every value in the data. It is the balance point of the data: if the values were equal weights on a ruler, the ruler would balance at the mean. The deviations from the mean, $x - \bar{x}$, always add up to zero, which is why the mean lies between the extremes and is pulled towards any value far from the rest.

The median

The median is the middle value once the data are in order. For $n$ values it is the $\frac{n+1}{2}$th value: when $n$ is odd this is a single value, and when $n$ is even it is halfway between the two middle values.

Half of the data lie on each side of the median. Only the order of the values matters, not their sizes, so changing the largest value from 25 to 2500 does not change the median at all.

The mode

The mode is the most frequent value. It is the only average that can be used for qualitative data (the most common blood group, the most popular colour), and it is useful when the most common value matters, such as the shoe size a shop should stock most of.

A data set with two modes is bimodal. If every value occurs once, there is no mode.

Data in a frequency table

When each value $x$ occurs with frequency $f$, the sum of all the data is $\sum fx$ and the number of data is $n = \sum f$. The mean is

where $k$ is the number of different values. The median is still the $\frac{n+1}{2}$th value, found by counting through the frequencies, and the mode is the value with the highest frequency.

In a study of 189 births, each mother was asked how many times she visited a physician during the first three months of pregnancy. The results are in the table.

Data: low birth weight study, Baystate Medical Center, Springfield, Massachusetts, 1986 (Hosmer and Lemeshow).

Grouped data: estimating the mean

When data are grouped into classes, the exact values are unknown. Each value is replaced by the mid-interval value of its class, which assumes the values in each class are spread evenly around its centre. The mean calculated this way is an estimate.

The class with the highest frequency is the modal class. For grouped data, the modal class takes the place of the mode, and the median is estimated from a cumulative frequency graph (see 4.2).

The waiting times between 272 eruptions of the Old Faithful geyser are grouped in the table, in classes of width 10 minutes.

Data: Old Faithful geyser, 272 eruptions (Härdle, 1991).

Solving for unknown values

Many questions give the mean and ask for a missing value. The key fact is that the sum of the data equals the mean multiplied by the number of values:

The same idea works for an unknown frequency: write $\sum fx$ and $\sum f$ in terms of the unknown, form an equation with the given mean, and solve.

For example, the table shows the number of goals, $x$, scored by a team in each match of a season, with one frequency unknown.

Combining two groups

The mean of two groups together is not the mean of the two means, unless the groups are the same size. The correct method combines the totals:

Quartiles of discrete data

The GDC method

Several slightly different rules for quartiles are used in different books and software. This course uses the rule built into a graphic display calculator (GDC):

1. Put the data in order and find the median.

2. Split the data into a lower half and an upper half. If $n$ is odd, leave the median out of both halves. If $n$ is even, the halves are the first $\frac{n}{2}$ and the last $\frac{n}{2}$ values.

3. $Q_1$ is the median of the lower half, and $Q_3$ is the median of the upper half.

For large data sets, the GDC's one-variable statistics give $Q_1$, the median and $Q_3$ directly, and these are the values to use.

For a frequency table, count through the frequencies to find the positions. For the 189 physician-visit values, the median is the 95th value (0). The lower half is the first 94 values, whose median is between the 47th and 48th values, both 0, so $Q_1 = 0$. The upper half is the 96th to 189th values; its middle values are the 142nd and 143rd values of the whole list, which lie among the values equal to 1 (positions 101 to 147), so $Q_3 = 1$.

Measures of dispersion

Why spread matters

A measure of dispersion describes how spread out the data are. Two data sets can have the same mean and still be very different. The data sets A: $4, 5, 6, 6, 7, 8$ and B: $1, 3, 6, 6, 9, 11$ both have mean 6 and median 6, but the values of B lie much further from the centre.

Range and interquartile range

The range is the simplest measure of spread.

The range is easy to find but depends only on the two most extreme values, so a single outlier can make it very large. The interquartile range, $\mathrm{IQR} = Q_3 - Q_1$, measures the spread of the middle half of the data and ignores the extremes (it is defined in 4.2). It is the natural partner of the median.

Variance and standard deviation

The range and IQR each use only two values. A measure that uses every value starts from the deviations $x - \mu$, the distances of the values from the mean. Their plain average is always zero, because positive and negative deviations cancel. Squaring them first makes every term positive, and averaging the squares gives the variance, $\sigma^2$.

The variance is measured in squared units (cm² for heights in cm), which is hard to interpret. Its square root, the standard deviation $\sigma$, is in the same units as the data, and is roughly the typical distance of a value from the mean.

A small standard deviation means the values cluster closely around the mean; a large one means they are widely scattered. A standard deviation of 0 means every value is the same. The standard deviation is never negative.

A second form of the variance

Expanding the squared deviation gives a form that is often quicker to use by hand. Since $\sum f = n$ and $\sum fx = n\mu$,

Dividing by $n$:

In words, the variance is the mean of the squares minus the square of the mean.

Standard deviation with a GDC

In practice the standard deviation is found with a GDC. Enter the data in a list (with a second list for frequencies, if there is a table), then choose one-variable statistics. The GDC shows two standard deviations:

The two values are close for large data sets but can differ noticeably for small ones.

Interpreting the standard deviation

A standard deviation is interpreted by comparison: with the mean of the same data (is the spread large relative to the typical value?) or with the standard deviation of another data set measured in the same units. A smaller standard deviation means more consistent values. An interpretation should always be written in the context of the data, not only as a number.

The effect of constant changes

Adding a constant

If the same number $b$ is added to every value, the whole distribution slides along by $b$ without changing shape. Every measure of centre moves by $b$, and every distance between values stays the same, so the measures of spread do not change.

Multiplying by a constant

If every value is multiplied by a positive number $a$, the whole distribution is stretched by a factor of $a$ about 0. Every measure of centre is multiplied by $a$, and every distance between values, including every deviation from the mean, is multiplied by $a$. The standard deviation, range and IQR are therefore multiplied by $a$, and the variance, which is built from squared deviations, by $a^2$.

The two effects combine. For a change $y = ax + b$ with $a > 0$:

If $a$ is negative, the order of the data is reversed but distances are still multiplied by $|a|$, so $\sigma_y = |a|\sigma_x$ and the variance is still multiplied by $a^2$. The mean is $a\bar{x} + b$ in every case.

Choosing an appropriate measure

Effect of outliers and skewness

The mean uses the size of every value, so a few very large or very small values pull it towards them. The median depends only on the order of the values, so it is resistant to outliers. In the same way, the standard deviation and range are strongly affected by outliers, while the IQR is not.

In a symmetric distribution, the mean and median are equal or close. In a skewed distribution they separate, and the mean lies further along the tail:

The river lengths are strongly positively skewed. Their mean, 591 miles, is larger than the lengths of two thirds of the rivers, so it does not describe a typical river well. The median, 425 miles, does. Similarly, the standard deviation, 492 miles, is inflated by the ten outliers, while the IQR, 378 miles, describes the spread of the typical rivers.

Guidelines

The mean and standard deviation are preferred when they are appropriate because they use every value and work well with further calculation (they are the parameters of the normal distribution in 4.9). The median and IQR are preferred when the data are skewed or contain outliers, which is why national statistics usually report median income rather than mean income.

The Old Faithful waiting times show the limits of any single average. Their mean (70.9 minutes) lies close to the trough between the two clusters of waits in the histogram in 4.2, where relatively few waits occur, and their median (76 minutes) sits at the lower edge of the larger cluster. Neither describes the short waits at all. A visitor is better told that the wait is usually either about 50 to 60 minutes or about 75 to 85 minutes.

Exam-style questions on measures of centre and spread

The birth weights of all 189 babies in the hospital study are grouped in the table.

Data: 189 births, Baystate Medical Center, Springfield, Massachusetts, 1986 (Hosmer and Lemeshow).

Self-check

• To do:

  I can calculate the mean, median and mode of a list of data and of a frequency table.

• To do:

  I can estimate the mean of grouped data using mid-interval values, and write down the modal class.

• To do:

  I can find the quartiles of discrete data using the GDC method, for odd and even numbers of values.

• To do:

  I can calculate the range, IQR, variance and standard deviation, by hand for small data sets and with a GDC using $\sigma_x$.

• To do:

  I can use $\sum x = n\bar{x}$ to find unknown values or frequencies, and combine the means of two groups.

• To do:

  I can state the effect of adding a constant, or multiplying by a constant, on each measure of centre and spread.

• To do:

  I can choose suitable measures of centre and spread for a data set, with reasons based on outliers and skewness.

1. (a)

   Find the lower quartile and the upper quartile of the recorded heights.

   [2]2 marks

   Markscheme (a)

   $Q_1 = 5.35$ A1

   $Q_3 = 7.15$ A1

2. (b)

   Show that $0.6$ and $11.8$ are both outliers.

   [2]2 marks

   Markscheme (b)

   $\text{IQR} = 7.15 - 5.35 = 1.8$ (A1)

   $Q_1 - 1.5 \times 1.8 = 2.65$

3. (c)

   Every seedling was at least $3$ cm tall when it was planted at the start of the three weeks. For each of the two outliers, state whether it should be kept in or removed from the data, giving a reason.

   [2]2 marks

   Markscheme (c)

   $0.6$ should be removed: a seedling at least 3 cm tall cannot have a height of 0.6 cm three weeks later, so it must be an error in recording (e.g. a misplaced decimal point or a value in different units) R1

   $11.8$ should be kept: it is possible for one seedling to grow much more than the others, so it is a valid (genuine) data value R1

   Note: Award R1 for each correct decision with a valid reason. Do not award R1 for a decision without a reason.

Upper half: $11, 12, 14, 17$, so $Q_3 = \frac{12 + 14}{2} = 13$.

$\mathrm{IQR} = 13 - 5.5 = 7.5$.

(b) $n = 10$ (even). The median is halfway between the 5th and 6th values: $\frac{28 + 30}{2} = 29$.

Lower half (first five values): $21, 23, 23, 26, 28$, so $Q_1 = 23$.

Upper half (last five values): $30, 31, 35, 36, 40$, so $Q_3 = 35$.

$\mathrm{IQR} = 35 - 23 = 12$.

(b) The means are almost the same, so the sprinters are equally fast on average. Sprinter A's standard deviation is about a third of sprinter B's, so A's times are much more consistent: all five are within 0.2 s of the mean, while B's vary from 9.9 s to 10.9 s. A is the better choice for a steady time under 10.5 s, although B is more likely to run the single fastest time.

Mean: apply the whole conversion.

Standard deviation: only the multiplication affects the spread; subtracting 32 does not.

Data: monthly mean air temperatures at Nottingham Castle, 1939 (Anderson, Time Series Analysis and Forecasting, 1976; the nottem data set in R).

[2]

(c) Dividing every weight by 1000 divides the mean and the standard deviation by 1000:

[2]

(d) $1 + 4 + 14 + 40 = 59$ babies weighed less than 2500 g. [1]

[1]

(e) $\mathrm{IQR} = 3515.5 - 2412 = 1103.5$ g. [1]

Lower boundary: $2412 - 1.5 \times 1103.5 = 2412 - 1655.25 = 756.75$ g. [1]

Since $709 < 756.75$, the weight of 709 g is an outlier. [1]

$0.6 < 2.65$ and $11.8 > 9.85$, so both are outliers AG