A frequency distribution condenses a mass of data, but a reader still wants a single representative value that summarises the entire distribution. This representative value, located at or near the centre of the distribution and around which all other observations tend to cluster, is called the average or the measure of central tendency. Averages condense the whole data into one figure, enable comparison between different groups, and serve as a basis for many statistical computations.
The commonly used measures of central tendency are the arithmetic mean, the median and the mode. Each of these has its own definition, methods of computation, merits and demerits, and suitability for particular types of data. The mean is computed from all observations and is the most widely used; the median is the positional average that divides the data into two equal halves and is best for open-end classes and skewed data; the mode is the value that occurs most frequently.
In this chapter we study the arithmetic mean for individual, discrete and continuous series; the median for the three types of series; the mode for different series; and the relationships between mean, median and mode, together with the relative merits, demerits and suitability of each average.
The arithmetic mean is the most common measure of central tendency. For individual observations, it is defined as the sum of all observations divided by the number of observations.
$$\text{Arithmetic Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}} = \frac{\sum X}{N}$$
where X represents the observations, N the number of observations, and the Greek letter sigma denotes summation.
For individual series, the mean is simply $\bar{X} = \frac{\sum X}{N}$. If a value is repeated, its frequency is taken into account.
For discrete series, the values are given with their frequencies. The mean is the weighted sum of the values, where the frequencies are the weights:
$$\bar{X} = \frac{\sum fX}{\sum f} = \frac{\sum fX}{N}$$
where f is the frequency of each value X, and N = sum of frequencies.
For continuous series, the class intervals are given with their frequencies. The mid-value of each class is taken as the representative value of that class. Then:
$$\bar{X} = \frac{\sum fM}{\sum f}$$
where M is the mid-value of each class = (Lower limit + Upper limit) / 2.
Short-cut method (assumed mean method): When the values are large, the mean can be computed conveniently by assuming an average (A) and taking deviations d = X - A.
$$\bar{X} = A + \frac{\sum fd}{\sum f}$$
Step-deviation method: When the class intervals are equal, deviations may be divided by the common class size h, taking d' = (X - A)/h, so that:
$$\bar{X} = A + \frac{\sum fd'}{\sum f} \times h$$
When different observations carry different degrees of importance, a simple mean would give equal weight to all, which may be misleading. In such cases we use the weighted arithmetic mean, where each observation is multiplied by its weight:
$$\bar{X}_w = \frac{\sum wX}{\sum w}$$
For example, when computing the average price of a basket of goods, the quantities consumed are used as weights, so that items consumed more heavily influence the average more.
The important properties of the arithmetic mean are:
The median is the value of the variable that divides the distribution into two equal parts, such that half of the observations lie above it and half below it. It is a positional average because its value depends on the position of observations in the ordered array, not on the magnitude of all observations.
For individual series: Arrange the data in ascending or descending order. If N is odd, the median is the value of the $\left(\frac{N+1}{2}\right)^{th}$ item. If N is even, the median is the average of the $\left(\frac{N}{2}\right)^{th}$ and $\left(\frac{N}{2}+1\right)^{th}$ items.
For discrete series: Compute the cumulative frequencies. The median is the value of the $\left(\frac{N+1}{2}\right)^{th}$ item, located from the cumulative frequencies.
For continuous series: The median class is the class in which the $\frac{N}{2}^{th}$ item lies. The median is then given by:
$$\text{Median} = L + \frac{\frac{N}{2} - cf}{f} \times h$$
where L is the lower limit of the median class, N is the total frequency, cf is the cumulative frequency of the class preceding the median class, f is the frequency of the median class, and h is the class size.
The median is not affected by extreme values, is easy to understand, and can be computed even with open-end classes. It is also the basis for the quartiles and percentiles used in dispersion analysis.
The mode is the value of the variable that occurs most frequently in the data. It is the most typical value in the sense of being the most common. A distribution may have one mode (unimodal), two modes (bimodal) or more (multimodal).
For individual series: The value that occurs the maximum number of times is the mode.
For discrete series: The value with the highest frequency is the mode.
For continuous series: The modal class is the class with the highest frequency. The mode is then computed by the formula:
$$\text{Mode} = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$$
where L is the lower limit of the modal class, f1 is the frequency of the modal class, f0 is the frequency of the class preceding the modal class, f2 is the frequency of the class following the modal class, and h is the class size.
Alternatively, the mode can be obtained from the empirical relationship:
$$\text{Mode} = 3 \text{ Median} - 2 \text{ Mean}$$
This relationship holds approximately for moderately skewed distributions.
For a symmetric distribution, mean, median and mode coincide. For a positively skewed distribution (a few extremely large values), the mean is greater than the median, which is greater than the mode. For a negatively skewed distribution, the order is reversed: Mean < Median < Mode.
The empirical relationship among the three averages is:
$$\text{Mode} = 3 \text{ Median} - 2 \text{ Mean}$$
This relationship is used to estimate one average when the other two are known.
Arithmetic mean: Merits - it is based on all observations, is rigidly defined, easy to calculate, and has useful mathematical properties. Demerits - it is unduly affected by extreme values, cannot be computed for open-end classes, and cannot be located graphically.
Median: Merits - it is not affected by extreme values, can be computed even with open-end classes, can be determined graphically from the ogive, and is easy to understand. Demerits - it is not based on all observations, is not amenable to algebraic treatment, and is affected by sampling fluctuations more than the mean.
Mode: Merits - it is easy to understand, is not affected by extreme values, and is useful in business where the most common size or value matters. Demerits - it is not rigidly defined (a distribution may have two modes), is not based on all observations, and is unstable for small samples.
| Measure | Formula (Individual) | Formula (Continuous) |
|---|---|---|
| Arithmetic mean | Sum X / N | Sum fM / Sum f |
| Short-cut mean | - | A + Sum fd / Sum f |
| Step-deviation | - | A + (Sum fd' / Sum f) x h |
| Median | (N+1)/2 th item | L + [(N/2 - cf)/f] x h |
| Mode | Most frequent value | L + [(f1-f0)/(2f1-f0-f2)] x h |
| Average | Best Suited For | Major Limitation |
|---|---|---|
| Arithmetic mean | Symmetric data, further algebra | Affected by extreme values |
| Median | Open-end classes, skewed data | Not based on all values |
| Mode | Most common value needed | Not rigidly defined |
| Weighted mean | Data with unequal importance | Requires knowledge of weights |
This chapter covered the three principal measures of central tendency - the arithmetic mean, the median and the mode - together with the weighted mean and the relationships among them. We learned to compute the mean for individual, discrete and continuous series, using the direct, short-cut and step-deviation methods, and studied its important properties and its sensitivity to extreme values. We defined the median as a positional average, computed it for all three types of series, and located it graphically at the intersection of the two ogives. We defined the mode as the most frequently occurring value and applied the empirical relation Mode = 3 Median - 2 Mean. Finally, we compared the merits and demerits of the three averages and noted their suitability for different situations. Central tendency alone, however, does not fully describe a distribution; two distributions with the same mean can be very different in their spread, which brings us to the study of measures of dispersion in the next chapter.