68
The arithmetic mean of a data set
Textbook: Part 2, lesson 68, pp. 130–133
GoalThe arithmetic mean of numbers; average speed.
New words
arithmetic mean · o‘rta arifmetikdata series · ma’lumotlar qatoriaverage speed · o‘rtacha tezliksum · yig‘indi
Explanation
The arithmetic mean of several numbers is the sum of the numbers divided by how many there are. For example, the mean of 7, 12 and 11 is (7 + 12 + 11) : 3 = 10. The mean of two numbers lies exactly in the middle of them on the number ray: for 4 and 12 it is 8. The mean is always greater than the smallest number and less than the largest (if the numbers are not all equal). Average speed is the whole distance divided by the whole time: v = S : t. It is calculated from the distance and time, not just by averaging the speeds of the parts.
Worked examples
Pages read in three weeks: 12, 15 and 9. The sum is 36, the mean 36 : 3 = 12 pages.
A bus drove 3 hours at 60 km/h and 2 hours at 70 km/h. Distance: 60 · 3 + 70 · 2 = 180 + 140 = 320 km; time 5 hours; average speed 320 : 5 = 64 km/h.
Class activity
“Class average”: ask 6 classmates for their step length or height (cm). Calculate the mean and check that it lies between the smallest and the largest value.
Practice
1
Find the arithmetic mean of 8, 12, 16 and 20.
14
2
A train went 2 hours at 50 km/h and 3 hours at 60 km/h. Its average speed (km/h)?
56
3
The mean of 5 numbers is 8. What is their sum?
40
4
Why does the mean lie between the smallest and the largest number?
Small numbers pull the mean down and large ones pull it up; it cannot exceed the largest or fall below the smallest.