30
Mean, mode and median
Textbook: pp. 194–199
GoalFind the mean, weighted mean, mode, median and range of a sample.
New words
mean · o‘rta qiymatmode · modamedian · medianarange · tanlanma kengligi
Explanation
The mean of a sample is x̄ = (x₁ + x₂ + … + xₙ)/n. If variant x₁ occurs n₁ times, x₂ occurs n₂ times and so on, the weighted mean is x̄ = (n₁x₁ + n₂x₂ + …)/(n₁ + n₂ + …). The mode is the value that occurs most often; a sample may have no mode or several. The median is the middle term of the variation series: for an odd number of terms it is the middle term, for an even number it is the mean of the two middle terms. The range is the difference between the largest and smallest variants and shows how spread out the data are.
Worked examples
3, 6, 6, 7, 8, 9, 10: mean (3 + 6 + 6 + 7 + 8 + 9 + 10) : 7 = 49 : 7 = 7; mode 6; median (4th term) 7; range 10 - 3 = 7.
A shop sold 20 kg of apples at 12 000 so‘m per kg and 30 kg at 15 000 so‘m per kg. The average price of 1 kg (weighted mean): (20 · 12 000 + 30 · 15 000) : 50 = 690 000 : 50 = 13 800 so‘m. The median of the even-sized series 2, 3, 6, 9, 10, 13 is (6 + 9) : 2 = 7.5.
Class activity
“Class statistics”: pupils measure their heights; groups find the mean, mode, median and range and discuss which indicator describes the class best.
Practice
1
Find the mean of 4, 8, 9, 11, 13.
9
2
Find the mode and median of the sample 2, 5, 5, 7, 5, 2.
mode 5, median 5
3
Find the range of the sample 12, 7, 19, 3, 10.
16
4
Why do the mean and median of 1, 2, 3, 100 differ so much?
The very large value 100 pulls the mean up (26.5), while the median (2.5) depends on the middle terms and is not sensitive to an extreme value.