The Mean
The mean is the value each observation would have if the total were shared equally. This picture gives the formula a meaning before any notation is used.
Mean means equal share
original values
2 + 4 + 6 + 8 = 20
share the same total
20 ÷ 4 values = 5
mean
Each equal share is 5
Build the total, then divide
The capital Greek letter means “add”. Therefore means “add every -value”.
Symbols
- = mean (same unit as x)
- = sum of all data values (same unit as x)
- = number of values (none)
A frequency table records the same value several times. Multiplying by its frequency rebuilds that part of the total.
Worked example
Mean from a frequency table
Values 2, 4 and 7 have frequencies 3, 2 and 5. The total is and the count is 10.
A frequency is a multiplier. Always check that is the total number of observations.
Grouped data gives an estimated mean
A midpoint stands in for every hidden value
class 0 ≤ x < 10
midpoint = 5
class 10 ≤ x < 20
midpoint = 15
class 20 ≤ x < 36
midpoint = 28
Use midpoint × frequency, so the grouped mean is estimated.
The raw values inside a class are unknown. Use the class midpoint as a representative value for every observation in that class. Write the midpoint as . The symbol means “approximately equal to”.
- Find every midpoint. Average the lower and upper class boundaries.
- Multiply midpoint by frequency. This estimates the total contributed by that class.
- Add and divide. Divide by and call the result an estimate.
Examiner note
Common mistake
Now you try
Times, in minutes, are recorded to the nearest minute for 240 competitors:
1–10: 12, 11–20: 38, 21–25: 68,
26–30: 76, 31–50: 46.
Estimate the mean time.
Show worked answer
The midpoints are 5.5, 15.5, 23, 28 and 40.5. Their weighted total is .