Central Tendency (Mean, Median, Mode)
Three ways to describe the centre of data. Always sort before finding median or mode.
A. What is Central Tendency?
Central tendency is a single number that represents the centre or typical value of a dataset. There are three measures: Mean, Median, and Mode.
Mean (Average)
Add all values together, then divide by how many values there are.
FormulaMean = Sum of all values ÷ Number of values
Median (Middle value)
The middle value when data is arranged in order. If there are two middle values, average them.
Formula
Odd count → middle value
Even count → average of the two middle values
Even count → average of the two middle values
Mode (Most frequent)
The value that appears most often. A dataset can have no mode, one mode, or multiple modes.
Example
2, 3, 3, 5, 7 → Mode = 3 (appears twice)
1, 2, 3, 4 → No mode (all appear once)
1, 2, 3, 4 → No mode (all appear once)
Range
The spread of the data — difference between the highest and lowest value.
FormulaRange = Maximum − Minimum
B. Finding Mean, Median, Mode
Where Mean, Median & Mode Sit on the Data
Full Example — Data: 4, 7, 3, 9, 7, 5, 7, 2, 6, 10
1Sort first: 2, 3, 4, 5, 6, 7, 7, 7, 9, 10
2Mean: Sum = 2+3+4+5+6+7+7+7+9+10 = 60 → 60 ÷ 10 = 6
3Median: 10 values (even) → average of 5th and 6th = (6+7) ÷ 2 = 6.5
4Mode: 7 appears 3 times → 7
5Range: 10 − 2 = 8
Mean=6 Median=6.5 Mode=7 Range=8
⚠ Always sort the data first before finding median or mode. Students who skip this step get the wrong median every time.
C. Mean from Frequency Table
Frequency Table Mean
When data is given in a table with frequencies, multiply each value by its frequency, add all those up, then divide by total frequency.
FormulaMean = Σ(f × x) ÷ Σf where f = frequency, x = value
Example — Find the mean from this frequency table
1Value 10, f=3 → f×x = 30
2Value 20, f=5 → f×x = 100
3Value 30, f=2 → f×x = 60
4Σf = 3+5+2 = 10 Σfx = 30+100+60 = 190
5Mean = 190 ÷ 10 = 19
Mean = 19
Example 2 — When to use each measure
?Data: 2, 3, 4, 5, 100 — which measure best represents the data?
1Mean = (2+3+4+5+100) ÷ 5 = 114 ÷ 5 = 22.8 — distorted by 100
2Median = 4 — middle value, not affected by 100
Median (4) better represents this data — 100 is an outlier
When to use which: Mean — when data has no extreme outliers. Median — when data has one very large or very small outlier. Mode — for non-numeric or categorical data.
Quick MCQ Revision
| Measure | How to find it | Best used when |
|---|---|---|
| Mean | Sum ÷ count | No extreme outliers |
| Median | Middle value (sort first!) | Outliers present |
| Mode | Most frequent value | Categorical data |
| Range | Max − Min | Measuring spread |
| Freq. Mean | Σ(fx) ÷ Σf | Data in frequency table |