60-second revision
- Mean = sum ÷ number of values; with frequencies, sum of fx ÷ sum of f.
- Median: sort first; for even n, average the two middle values.
- Mode = most frequent value; data can have more than one mode.
- Empirical relation: Mode = 3 Median − 2 Mean.
- Combined mean = (n1 m1 + n2 m2)/(n1 + n2).
- Corrected mean = old mean + (correct value − wrong value)/n.
- Adding a constant shifts mean, median and mode but leaves range unchanged.
Mean (arithmetic average)
Mean = sum of observations ÷ number of observations. For a frequency table, mean = sum of (f × x) ÷ sum of f.
| Series | Mean |
|---|---|
| First n natural numbers | (n + 1)/2 |
| First n odd numbers | n |
| First n even numbers | n + 1 |
| Any arithmetic progression | (first + last)/2 |
Worked example: daily sales of 4, 6, 6, 9 and 10 units have mean 35/5 = 7.
Worked example (weighted mean): a test has 3 sections with weights 2, 3 and 5. A student scores 60, 70 and 80. Weighted mean = (120 + 210 + 400)/10 = 73, not the simple average 70.
Combined, corrected and changed means
- Combined mean of two groups = (n1 × m1 + n2 × m2)/(n1 + n2). Example: 40 boys averaging 52 and 60 girls averaging 62 give (2080 + 3720)/100 = 58.
- Wrong value copied: correct mean = old mean + (correct − wrong)/n. Example: mean of 25 values is 30 but 46 was read as 21; the correct mean is 30 + 25/25 = 31.
- Value removed: removed value = n × old mean − (n − 1) × new mean. Example: 8 numbers average 15; dropping one makes it 14, so the dropped number is 120 − 98 = 22.
- Value added: new value = (n + 1) × new mean − n × old mean.
Median (the middle value)
Arrange the data in order first. For n values: if n is odd, the median is the ((n + 1)/2)th value; if n is even, it is the average of the (n/2)th and (n/2 + 1)th values.
Worked example: 11, 4, 7, 19, 2 sorted is 2, 4, 7, 11, 19, so the median is 7. For 3, 9, 6, 12, sorted 3, 6, 9, 12, the median is (6 + 9)/2 = 7.5.
Grouped data: median = L + ((N/2 − cf)/f) × h, where L is the lower limit of the median class, cf the cumulative frequency before it, f its frequency and h the class width. Worked example: L = 20, N = 50, cf = 18, f = 14, h = 10 gives 20 + (7/14) × 10 = 25.
Mode and range
The mode is the value that occurs most often. A set can have one mode, two (bimodal) or more. Range = highest value − lowest value, the simplest measure of spread.
Empirical relation for moderately skewed data: Mode = 3 Median − 2 Mean. Worked example: mean 45 and median 42 give mode = 126 − 90 = 36.
What changes when you transform data
| Operation on every value | Mean, median, mode | Range, standard deviation |
|---|---|---|
| Add a constant k | Each increases by k | Unchanged |
| Subtract a constant k | Each decreases by k | Unchanged |
| Multiply by k | Each multiplied by k | Multiplied by |k| |
| Divide by k | Each divided by k | Divided by |k| |
Worked example: if values with mean 12 and range 9 are each doubled and then increased by 3, the new mean is 27 and the new range is 18.
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