Mean, Median and Mode

Introduction to Mean, Median and Mode

Mean, median, and mode are three measures of central tendency — statistical values that attempt to summarise a whole set of data using a single, representative number. Each measure describes the "centre" or "typical value" of a data set in a slightly different way, and understanding when and how to use each one is an essential statistical skill used in analysing test scores, sports statistics, financial data, and countless other real-world data sets.

Meaning of the Mean

The mean (often simply called the "average") is calculated by adding together all the values in a data set and dividing the total by the number of values. The formula is: Mean = (Sum of all values) ÷ (Number of values). For example, to find the mean of the data set 4, 8, 6, 10, 2: Sum = 4 + 8 + 6 + 10 + 2 = 30, and there are 5 values, so Mean = 30 ÷ 5 = 6.

Properties of the Mean

The mean takes into account every single value in the data set, making it sensitive to all the data, but this also means it can be significantly affected by unusually high or low values, called outliers. For example, in the data set 2, 3, 4, 5, 50, the mean is (2+3+4+5+50) ÷ 5 = 64 ÷ 5 = 12.8, a value much higher than most of the data points, because it has been pulled upward by the outlier, 50.

Meaning of the Median

The median is the middle value of a data set when the values are arranged in order (either ascending or descending). To find the median: first arrange all the data values in order from smallest to largest; if there is an odd number of values, the median is the single middle value; if there is an even number of values, the median is the mean (average) of the two middle values. For example, for the data set 3, 7, 9, 12, 15 (already ordered, 5 values), the median is the middle value, 9. For the data set 4, 8, 10, 14 (4 values), the median is the mean of the two middle values, (8+10) ÷ 2 = 9.

Properties of the Median

Unlike the mean, the median is not significantly affected by outliers, since it depends only on the position of values in an ordered list, not their actual size. This makes the median a particularly useful measure of central tendency when a data set contains extreme values that might distort the mean. For example, for the data set 2, 3, 4, 5, 50, the median is 4 (the middle value), which is a much more typical representation of most of the data than the mean of 12.8.

Meaning of the Mode

The mode is the value that appears most frequently in a data set. A data set can have one mode (unimodal), more than one mode (bimodal for two modes, or multimodal for more), or no mode at all if every value appears the same number of times. For example, in the data set 2, 3, 3, 5, 7, 7, 7, 9, the mode is 7, since it appears three times, more than any other value.

Properties of the Mode

The mode is the only measure of central tendency that can be used with non-numerical (categorical) data, such as favourite colours or subjects, since it simply identifies the most common category rather than requiring any calculation. However, the mode may not always exist, or a data set may have multiple modes, which can make it less useful as a single summary value in some situations, especially when data values are widely spread out with few exact repetitions.

Choosing the Appropriate Measure

The mean is generally the most commonly used and mathematically useful measure, especially for further statistical calculations, but is best avoided when a data set contains significant outliers. The median is more appropriate when a data set contains outliers or is heavily skewed (unevenly distributed), since it is not distorted by extreme values. The mode is most useful for categorical data, or when identifying the single most common or popular value or category is specifically important, such as the most popular shoe size sold by a shop.

Finding the Mean from a Frequency Table

When data is presented in a frequency table, the mean is calculated by multiplying each value by its frequency, summing these products, and dividing by the total frequency (the total number of data values). The formula is: Mean = Σ(fx) ÷ Σf, where f represents frequency, x represents each data value, and Σ means "the sum of." For example, if 3 students scored 5, 5 students scored 7, and 2 students scored 9 on a quiz, the mean is calculated as: ((3×5) + (5×7) + (2×9)) ÷ (3+5+2) = (15+35+18) ÷ 10 = 68 ÷ 10 = 6.8.

The Range

Although not itself a measure of central tendency, the range is often studied alongside mean, median, and mode, as a simple measure of how spread out a data set is. The range is calculated as: Range = Highest value − Lowest value. For example, for the data set 4, 8, 6, 10, 2, the range is 10 − 2 = 8. A larger range indicates more variability (spread) in the data, while a smaller range indicates that the data values are closer together.

Comparing Data Sets Using Central Tendency

Mean, median, mode, and range are often used together to compare two or more data sets, such as comparing the exam performance of two different classes. For example, if one class has a higher mean score than another, it generally performed better on average, but comparing the range of each class also reveals whether one class had more consistent performance (a smaller range) or more varied performance (a larger range) than the other.

Real-Life Applications

Measures of central tendency are used extensively in real life: calculating average scores, grades, or performance ratings in schools and workplaces; summarising typical income, prices, or expenses in economics and business; analysing sports statistics, such as average goals scored or typical player performance; and interpreting scientific or medical data, such as average recovery times or typical measurements in research studies.

Common Mistakes with Mean, Median and Mode

Common errors include forgetting to arrange data in order before finding the median, incorrectly calculating the median for an even number of values (forgetting to average the two middle values), confusing which value is the mode when multiple values have similar frequencies, and using the mean when the median would be more appropriate for data containing significant outliers.

Summary

Mean, median, and mode are three measures of central tendency that summarise a data set using a single representative value. The mean is the sum of all values divided by the number of values, the median is the middle value when data is arranged in order, and the mode is the most frequently occurring value. Each measure has its own strengths and is more appropriate in different situations, particularly regarding how sensitive each measure is to outliers, and together with the range, they provide valuable tools for summarising, comparing, and interpreting data sets.

Worked Examples

Example 1: Find the mean of 12, 15, 9, 18, 6. Sum = 12+15+9+18+6 = 60. Mean = 60 ÷ 5 = 12.

Example 2: Find the median of 7, 3, 9, 5, 11. Arranged in order: 3, 5, 7, 9, 11. The middle value is 7.

Example 3: Find the mode of 4, 7, 7, 9, 7, 2, 9. The value 7 appears three times, more than any other value, so the mode is 7.

Example 4: Find the range of 14, 8, 20, 5, 17. Range = highest − lowest = 20 − 5 = 15.

Example 5: In a class, 5 students scored 6, 10 students scored 8, and 5 students scored 10 on a quiz. Find the mean score. Mean = Σ(fx) ÷ Σf = ((5×6)+(10×8)+(5×10)) ÷ (5+10+5) = (30+80+50) ÷ 20 = 160 ÷ 20 = 8.

Student Exercise

Solve the following problems, showing all your working:

1. Find the mean of 20, 25, 15, 30, and 10.
2. Find the median of 6, 2, 9, 4, 11, and 8.
3. Find the mode of 3, 3, 5, 7, 7, 7, and 9.
4. Find the range of 45, 12, 38, 7, and 29.
5. In a class, 2 students scored 5, 4 students scored 7, and 4 students scored 9 on a test. Find the mean score.
6. Find the median of 15, 22, 18, 9, 30, 25, and 12.
7. Find the mean of 3, 6, 9, 12, and 15.
8. State whether the data set 2, 4, 4, 6, 8, 8 is unimodal or bimodal, and give the mode(s).
9. Class A has a mean score of 68 and a range of 20. Class B has a mean score of 71 and a range of 35. Which class performed more consistently, and why?
10. In a class, 3 students scored 4, 5 students scored 6, and 2 students scored 8 on a quiz. Find the mean score.