Mean, Median and Mode: When to Use Each

Mean, median and mode are the three most common ways to describe the "centre" of a data set. Each one answers a slightly different question, and choosing the wrong one can give a misleading picture. This guide shows you how to calculate all three and, more importantly, when to use each.

The Three Measures of Central Tendency

Mean

The arithmetic mean is the sum of all values divided by the number of values:

$$ \bar{x} = \frac{\sum x}{n} $$

For a frequency distribution, use

$$ \bar{x} = \frac{\sum fx}{\sum f} $$

The mean uses every value in the data, which is its strength and its weakness.

Median

The median is the middle value when the data are arranged in order.

• If \( n \) is odd, the median is the \( \left(\frac{n+1}{2}\right) \)th value.
• If \( n \) is even, it is the average of the \( \frac{n}{2} \)th and \( \left(\frac{n}{2} + 1\right) \)th values.

Mode

The mode is the value that occurs most often. A data set can have one mode (unimodal), two modes (bimodal), several modes, or no mode at all if every value appears equally often.

Worked Example 1: Monthly Salaries

A small office has seven employees with the following monthly salaries (in thousands of rupees):

$$ 40,\; 45,\; 45,\; 50,\; 55,\; 60,\; 300 $$

The 300 belongs to the owner.

Mean:

$$ \sum x = 40 + 45 + 45 + 50 + 55 + 60 + 300 = 595 $$

$$ \bar{x} = \frac{595}{7} = 85 $$

Median: The data are already in order and \( n = 7 \), so the median is the \( \frac{7+1}{2} = 4 \)th value, which is 50.

Mode: 45 appears twice; every other value appears once. The mode is 45.

Interpretation: A mean of 85 thousand suggests a typical employee earns 85 thousand, but six of the seven people earn 60 thousand or less. The single extreme value (an outlier) pulls the mean upwards. The median of 50 thousand is a far better description of a typical salary.

What if we remove the outlier? The remaining six values sum to \( 595 - 300 = 295 \), so the mean drops to \( 295 / 6 \approx 49.17 \). The median becomes the average of the 3rd and 4th values, \( (45 + 50)/2 = 47.5 \). The mean changed by about 36 thousand; the median changed by only 2.5 thousand. This shows how robust the median is.

Worked Example 2: Grouped Frequency Data

The marks of 40 students in a test are shown below.

Step 1: Estimated mean.

$$ \sum f = 4 + 10 + 16 + 10 = 40, \qquad \sum fx = 20 + 150 + 400 + 350 = 920 $$

$$ \bar{x} = \frac{920}{40} = 23 $$

Step 2: Median. The median position is \( \frac{n}{2} = 20 \). Cumulative frequencies are 4, 14, 30, 40, so the 20th value lies in the class 20–30. Using the formula

$$ \text{Median} = L + \frac{\frac{n}{2} - F}{f}\times h $$

with lower boundary \( L = 20 \), cumulative frequency before the class \( F = 14 \), class frequency \( f = 16 \) and width \( h = 10 \):

$$ \text{Median} = 20 + \frac{20 - 14}{16}\times 10 = 20 + 3.75 = 23.75 $$

Step 3: Modal class and mode. The highest frequency (16) is in the class 20–30, so this is the modal class. Using

$$ \text{Mode} = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2}\times h $$

with \( f_1 = 16 \), \( f_0 = 10 \) (class before) and \( f_2 = 10 \) (class after):

$$ \text{Mode} = 20 + \frac{16 - 10}{32 - 10 - 10}\times 10 = 20 + \frac{6}{12}\times 10 = 25 $$

All three measures are close together (23, 23.75 and 25), which indicates the distribution is roughly symmetric.

When to Use Each Measure

Use the Mean When

• The data are numerical and roughly symmetric.
• There are no extreme outliers.
• You need a value for further calculations, such as variance or standard deviation.

Examples: average height of students, average daily temperature.

Use the Median When

• The data are skewed or contain outliers.
• The data are ordinal (ranked) rather than truly numerical.

Examples: household income, house prices, waiting times.

Use the Mode When

• The data are categorical (names, colours, brands), where the mean and median make no sense.
• You want the most common or most popular value.

Examples: most popular shoe size, most common blood group, favourite subject.

Mean, Median, Mode and Skewness

The relationship between the three measures reveals the shape of a distribution:

• Symmetric: mean ≈ median ≈ mode.
• Positively (right) skewed: mean > median > mode, as in the salary example.
• Negatively (left) skewed: mean < median < mode.

For moderately skewed data, the empirical relation \( \text{Mode} \approx 3\,\text{Median} - 2\,\text{Mean} \) is often used as an estimate.

Common Mistakes

• Forgetting to arrange data in order before finding the median.
• Using class limits instead of midpoints when calculating the mean of grouped data.
• Choosing the mean for income or price data that contain extreme values.
• Thinking a data set must have exactly one mode.
• Calling the grouped-data mean "exact"; it is only an estimate because midpoints are used.

Key Takeaways

• The mean uses every value but is sensitive to outliers.
• The median is the middle value and is robust, ideal for skewed data.
• The mode is the most frequent value and the only average for categorical data.
• Comparing the three reveals whether a distribution is symmetric or skewed.

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