Statistics: Measures of Central Tendency and Dispersion

Mean, median and mode

Mean, median and mode are the three principal measures of central tendency. Each names a single value that locates the centre of a data set, but each is defined differently, so for the same data they generally take different values.

The arithmetic mean of n observations is \bar{x}=\dfrac{\sum x_i}{n}. It uses every observation, so changing any one value changes the mean. If the observations are measured from a convenient number a, then \bar{x}=a+\dfrac{1}{n}\sum (x_i-a). Hence the total of the deviations from a fixes how far the mean lies from a. When the sums of deviations from two different numbers are known, subtracting the two relations gives the number of observations and then the total. For an arithmetic progression with first term a and last term l, the mean equals \dfrac{a+l}{2}, because the terms are symmetric about the centre.

The median is the middle value after the observations are arranged in order. For n odd it is the \dfrac{n+1}{2}th observation; for n even it is the average of the \dfrac{n}{2}th and \left(\dfrac{n}{2}+1\right)th observations. For an arithmetic progression with an odd number of terms, the median is the middle term and coincides with the mean. In a discrete frequency table, find the cumulative frequencies and locate the value at which the cumulative frequency first reaches \dfrac{N}{2}, or the two central positions when N is even. The median of a discrete distribution is always one of the observed values.

For grouped data, the median class is the first class whose cumulative frequency reaches \dfrac{N}{2}, and $M = L + \frac{\frac{N}{2}-cf}{f}\times h$ where L is the lower limit of the median class, cf the cumulative frequency before it, f its frequency and h its width. The median is obtained by linear interpolation inside that class.

The mode is the value with the greatest frequency. A data set may have no mode, one mode, or several. A value is the unique mode only if it occurs strictly more often than every other value; if two or more values share the highest frequency, there is no unique mode. For grouped data, the modal class is the class with the highest frequency, and $Z = L + \frac{f_1-f_0}{2f_1-f_0-f_2}\times h$ where f_0 and f_2 are the frequencies of the classes just before and after the modal class.

The three measures are related through the shape of the distribution. In a symmetric unimodal distribution they coincide. If the tail of a unimodal distribution points to the left (negative skew), the mean is pulled toward the tail, so \text{Mode} > \text{Median} > \text{Mean}; for a tail to the right the order reverses. A single large or small value pulls the mean but barely moves the median or mode, so a set of consecutive-type values with a shifted top value can have mean greater than median while the mode stays at the repeated value. Problems that give proportional relations such as 5P=4Q=\dfrac{R}{2} are solved by writing P, Q, R as multiples of one parameter t; the required ratio then becomes free of t.

Key points:

• Mean — sum divided by count; uses every value.
• Median — middle value of ordered data; for grouped data use M=L+\frac{N/2-cf}{f}h.
• Mode — most frequent value; must be strictly most frequent to be unique.

Mean of grouped frequency distribution

For a frequency distribution, the mean is the weighted average \bar{x}=\dfrac{\sum f_i x_i}{\sum f_i}, where x_i is a value (discrete table) or a class midpoint (class intervals) and f_i its frequency. For a class interval the midpoint is the average of its two limits, and the midpoints are then used exactly as discrete values.

Multiplying every frequency by the same positive constant leaves the mean unchanged, because the constant appears in both the numerator \sum f_i x_i and the denominator \sum f_i and cancels. Only the total frequency changes.

Binomial coefficients often serve as frequencies. Two standard identities are \sum_{k=0}^{n}\binom{n}{k}=2^n and \sum_{k=0}^{n}k\binom{n}{k}=n\,2^{n-1}. If the value k+1 carries frequency \binom{n}{k}, the total frequency is 2^n and the weighted sum is n2^{n-1}+2^n, so the mean is \dfrac{n}{2}+1. The identity for \sum k\binom{n}{k} is derived from k\binom{n}{k}=n\binom{n-1}{k-1}.

When the frequencies form a geometric progression, the sums \sum f_i and \sum x_i f_i are finite geometric-type sums. For example, if x_i = i and f_i = r^{i-1}, then \sum_{i=1}^{n} i r^{i-1}=\dfrac{1-(n+1)r^n+nr^{n+1}}{(1-r)^2} for r\ne 1. Substituting r=\tfrac12 gives the weighted sum, and dividing by \sum f_i = \dfrac{1-r^n}{1-r} gives the mean. Checking small values of n (such as n=1 and n=2) is a reliable way to confirm the closed form.

For a discrete frequency table, the median is found from cumulative frequencies. When N is even, the median is the average of the \frac{N}{2}th and (\frac{N}{2}+1)th observations; if both positions lie within the same cumulative group, the median is that value.

Key points:

• Weighted mean — \sum f_i x_i / \sum f_i.
• Class midpoint — (\text{lower limit}+\text{upper limit})/2.
• Scaling frequencies — the mean does not change.

Variance of raw data

The variance of n observations is the mean of the squared deviations from the mean, \sigma^2=\dfrac{1}{n}\sum (x_i-\bar{x})^2. Its computational form is $\sigma^2=\frac{\sum x_i^2}{n}-\bar{x}^2 = \frac{\sum x_i^2}{n}-\left(\frac{\sum x_i}{n}\right)^2.$ When the totals \sum x_i and \sum x_i^2 are given, this form avoids computing each deviation.

The standard deviation is \sigma=\sqrt{\sigma^2}, which is expressed in the same unit as the data. Because the variance averages squared deviations, its unit is the square of the data unit. Given the variance, the standard deviation follows by taking the non-negative square root.

When some observations are unknown but the mean and variance are given, write the sum and sum of squares as equations in the unknowns. The mean gives the total, and the variance gives the total of squares. Eliminating one unknown then produces a quadratic in the other. For positive observations, only the root that is positive should be kept, and it must also satisfy the original equations. With two unknowns, the sum gives one linear relation and the sum of squares gives another; the product of the two unknowns follows from (x+y)^2-(x^2+y^2)=2xy, so the unknowns are the roots of a quadratic.

The variance of a subset of observations depends only on the chosen values. It must be computed afresh from the subset's own mean and squared deviations, and it cannot be inherited from the variance of the whole data.

Key points:

• Variance — \frac{1}{n}\sum (x_i-\bar x)^2 = \frac{\sum x_i^2}{n}-\bar x^2.
• Standard deviation — \sigma=\sqrt{\sigma^2}.
• Unknown observations — form the sum and sum-of-squares equations, then solve.

Deviations from mean and median

The sum of deviations of n values from a number a is \sum (x_i-a)=\sum x_i - na. This expression is the basis of most deviation problems. Taking a=\bar{x} gives \sum (x_i-\bar{x})=0, so the algebraic sum of deviations from the mean is always zero.

If the algebraic sum of deviations from an unknown number y is given, set \sum x_i - ny equal to that value and solve the resulting linear equation for y.

The sum of deviations from the median is not zero in general. It is zero only when the median coincides with the mean, as in a symmetric arithmetic progression with an odd number of terms. The usual exam trap is to assume that every 'deviation sum' vanishes.

When the same values are measured from two numbers a and b, the difference of the two deviation sums is \sum (x_i-a)-\sum (x_i-b)=n(b-a). Hence the difference of the sums determines n directly, and squaring it removes the sign.

The mean deviation about the median is \dfrac{1}{N}\sum f_i\,|x_i-M|, where M is the median. Absolute values are used, so the deviations do not cancel. The sum of absolute deviations is smallest when measured from the median, whereas the sum of squared deviations is smallest when measured from the mean.

Key points:

• Sum of deviations from the mean — always zero.
• Sum from a number a — \sum x_i - na.
• Mean deviation about median — \frac{1}{N}\sum f_i|x_i-M|.

Measures of dispersion

A measure of dispersion describes how far the observations spread about the centre. The range is the difference between the largest and smallest values. The quartile deviation is \dfrac{Q_3-Q_1}{2}. The mean deviation is \dfrac{1}{n}\sum |x_i-\bar{x}|, the variance is \sigma^2, and the standard deviation is \sigma=\sqrt{\sigma^2}. The range and the quartile deviation use only a few values, and so they ignore most of the data.

When the mean is the chosen measure of central tendency, the standard deviation is the most reliable and accurate measure of variability, because it uses every observation with squared deviations about the mean.

Adding the same constant to every observation shifts the mean by that constant but leaves every deviation from the mean unchanged. Therefore the standard deviation and variance are unaffected. Multiplying every observation by a constant c multiplies the standard deviation by |c| and the variance by c^2.

To compare variability between groups with different means, use the coefficient of variation CV=\dfrac{\sigma}{\bar{x}}\times 100. It expresses the standard deviation as a percentage of the mean. The group with the largest CV is the most variable relative to its average, which may differ from the group with the largest standard deviation.

The mean has the same unit as the observations. The mean deviation and the standard deviation also carry the unit of the data. The variance carries the square of that unit, so a statement that the mean and the variance share a unit is false.

Key points:

• Range — largest minus smallest value.
• Standard deviation — most reliable when the mean is the central measure.
• Coefficient of variation — \frac{\sigma}{\bar x}\times 100, used to compare variability.

Mean deviation about mean

The mean deviation (also called the mean absolute deviation) measures the average distance of the observations from a central value. When the mean is the central value, it is defined as \text{MD} = \frac{1}{N}\sum_{i=1}^{N} |x_i - \bar{x}|, where \bar{x} is the arithmetic mean. For grouped data with frequencies f_i and N = \sum f_i, the formula becomes \text{MD} = \frac{1}{N}\sum f_i |x_i - \bar{x}|, using class mid-values for x_i.

The absolute value is essential. Ordinary deviations from the mean always sum to zero, so without absolute values the average would carry no information. For a set of equally spaced values that is symmetric about its mean, the absolute deviations come in matching pairs, which makes the sum quick to compute.

The coefficient of mean deviation is the unit-free ratio \frac{\text{MD}}{\bar{x}}. It is used to compare the spread of two data sets with different means. For example, if a set has mean 30 and mean deviation 6, its coefficient is \frac{6}{30} = 0.2.

For a frequency table, first find the mean with \bar{x} = \frac{\sum f x}{\sum f}, then compute \sum f|x - \bar{x}| over the class mid-values and divide by N. Students often forget to weight each deviation by its frequency.

When the question asks for the mean deviation of a subset, such as the largest few observations of an ordered list, take only that subset. Its own mean must be computed, and the deviations are measured from that subset mean, not from the mean of the whole data.

Note that the sum of absolute deviations is minimized when measured from the median, not from the mean. The mean deviation about the mean is therefore usually slightly larger than the mean deviation about the median.