Measures of Central Tendency: Mean, Median, and Mode
Measures of central tendency summarise a dataset by identifying a typical or central value. The three main measures are mean, median, and mode, each with distinct calculation methods and uses.
The mean is calculated by adding all data values and dividing by the number of values. For example, for data 3, 5, 7, 9, 11: mean = (3 + 5 + 7 + 9 + 11) ÷ 5 = 35 ÷ 5 = 7.
The median is the middle value when data is arranged in order. For an odd number of values, it is the centre value; for an even number, it is the average of the two middle values. For example, data 4, 8, 12, 16, 20 has median 12; data 4, 8, 12, 16 has median (8 + 12) ÷ 2 = 10. The median is less affected by extreme values (outliers). For example, in data 2, 3, 4, 100, the mean is 27.25 but the median is 3.5, showing the median better represents the centre when outliers exist. The mode is the most frequently occurring value. For example, in data 2, 4, 4, 5, 7, the mode is 4. The mode is useful for categorical data or to identify common values.
- Mean: sum of all values ÷ number of values.
- Median: middle value in ordered data; average of two middle values if even number.
- Mode: most frequent value in the dataset.
Measures of Spread: Range, Quartiles, and Interquartile Range
Measures of spread describe how data values are distributed around the centre. The range is the simplest measure, calculated as the difference between the maximum and minimum values. For example, in data 3, 7, 8, 10, 15, range = 15 − 3 = 12.
Quartiles divide ordered data into four equal parts. The lower quartile (Q1) is the median of the lower half of data, the upper quartile (Q3) is the median of the upper half, and the median (Q2) is the middle quartile.
The interquartile range (IQR) is Q3 − Q1 and measures the spread of the middle 50% of data, reducing the effect of outliers. For example, for data 2, 4, 6, 8, 10, 12, 14: Q1 = 4, Q3 = 12, so IQR = 12 − 4 = 8.
- Range = maximum − minimum value.
- Quartiles split data into four equal parts.
- Interquartile range (IQR) = Q3 − Q1, spread of middle 50% of data.
Uses and Interpretation of Each Measure
The mean uses all data values and is sensitive to extreme values, making it suitable for symmetric distributions without outliers.
The median is robust to outliers and skewed data, providing a better centre measure when data is not symmetric.
The mode identifies the most common value, useful for categorical data or to highlight frequent occurrences in numerical data, such as the most common shoe size in a group of people or the modal class in grouped data (see later). The range gives a quick sense of total spread but is affected by outliers and does not show how data is distributed between extremes. Quartiles and the IQR provide more detailed information about data spread and are less influenced by extreme values, helping to understand variability and detect skewness.
- Mean: sensitive to all values, affected by outliers.
- Median: robust to outliers, better for skewed data.
- Mode: identifies common values, useful for categorical data.
- Range: simple spread measure, affected by extremes.
- Quartiles and IQR: describe spread of middle data, less affected by outliers.
Estimating the Mean from Grouped Data
When data is grouped into classes with frequencies, the exact values are unknown, so the mean is estimated using class midpoints.
For grouped discrete or continuous data, calculate the midpoint of each class (average of lower and upper class boundaries), multiply each midpoint by its class frequency, sum these products, then divide by the total frequency.
Example: For grouped data with classes 10–14 (frequency 3), 15–19 (frequency 5), 20–24 (frequency 2): midpoints are 12, 17, and 22 respectively. Estimated mean = (12×3 + 17×5 + 22×2) ÷ (3 + 5 + 2) = (36 + 85 + 44) ÷ 10 = 165 ÷ 10 = 16.5.
- Calculate class midpoints.
- Multiply midpoints by frequencies.
- Sum products and divide by total frequency.
- Result is estimated mean of grouped data.
Determining the Modal Class from a Grouped Frequency Distribution
The modal class is the class interval with the highest frequency in a grouped frequency distribution.
To find it, identify the class with the largest frequency value.
Example: For classes 5–9 (frequency 4), 10–14 (frequency 7), 15–19 (frequency 6), the modal class is 10–14 because 7 is the highest frequency.
- Identify class with highest frequency.
- That class is the modal class.
- Useful for understanding where data is most concentrated in grouped data.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Calculating Mean | Add all values, divide by number of values | Data: 2, 4, 6; Mean = (2+4+6)/3=4 | Show formula, substitution, and final answer clearly |
| Finding Median | Order data, find middle value (or average two middles) | Data: 3, 5, 7, 9; Median = (5+7)/2=6 | Order data carefully; state method for even/odd number |
| Identifying Mode | Find most frequent value | Data: 1, 2, 2, 3; Mode = 2 | State mode clearly; if no repeats, say 'no mode' |
| Estimating Mean from Grouped Data | Use class midpoints and frequencies | Classes: 10–14(3), 15–19(5); Mean ≈ (12×3 + 17×5)/8 | Calculate midpoints; multiply by frequency; divide sum by total frequency |
| Finding Modal Class | Identify class with highest frequency | Classes: 5–9(4), 10–14(7); Modal class = 10–14 | State modal class explicitly |
Active recall
Quick check
1. Calculate the mean of the data: 4, 8, 12, 16.
Mean = (4 + 8 + 12 + 16) ÷ 4 = 40 ÷ 4 = 10.
2. Find the median of the data: 3, 7, 9, 15, 21.
Data ordered: 3, 7, 9, 15, 21; median is middle value = 9.
3. Determine the mode of the data: 5, 5, 6, 7, 7, 7, 8.
Mode is 7, as it occurs most frequently (3 times).
4. Calculate the range of the data: 10, 14, 18, 22, 26.
Range = 26 − 10 = 16.
5. Estimate the mean from grouped data: Classes 10–14 (freq 4), 15–19 (freq 6).
Midpoints: 12 and 17; Estimated mean = (12×4 + 17×6) ÷ (4 + 6) = (48 + 102) ÷ 10 = 150 ÷ 10 = 15.