Count from the top or from the bottom
Less than counts up from the bottom; more than counts down from the total.
Twenty or more
We know that 8 of the 53 students scored less than 20. The rest scored 20 or more.
Each student is either below 20 or at 20 and above.
How many of the 53 students scored 20 or more?
What this lesson covers
The idea
A cumulative frequency distribution of the less than type gives the number of observations less than each upper class limit; one of the more than type gives the number more than or equal to each lower class limit.
Twenty or more
We know that 8 of the 53 students scored less than 20. The rest scored 20 or more.
Each student is either below 20 or at 20 and above.
How many of the 53 students scored 20 or more?
- 45
- 8
- 53
Take the classes off the total
The less than column is already filled. Open More than or equal to. All 53 students scored 0 or more. Tap each row: it takes off the students of the class above, because they scored less than this lower limit.
Two types of cumulative frequency
A less than distribution gives the number of observations less than each upper class limit. A more than distribution gives the number more than or equal to each lower class limit.
Less than: 10, 20, …, 100 are the upper limits, and the column counts up to 53. More than: 0, 10, …, 90 are the lower limits, and the column counts down from 53. At the same limit the two numbers add up to all the observations: for 40, 15 + 38 = 53.
Notes
Less than type: the number of observations less than each upper class limit. More than type: the number more than or equal to each lower class limit.
Check yourself
How many of the 53 students scored 30 or more?
Answer: 41
53 − 12 = 41. (Or take 45 and subtract the 4 students of 20–30: 45 − 4 = 41.)
NCERT Example 7: of 51 girls, 29 are shorter than 150 cm. How many are 150 cm or taller?
Answer: 22
51 − 29 = 22.
In Example 7, 11 girls are shorter than 145 cm and 29 are shorter than 150 cm. How many girls are in the class 145–150?
Answer: 18
29 − 11 = 18.
Which type of cumulative frequency is counted at the upper limits of the classes?
- Less than type — correct. Yes! "Less than 20" uses the upper limit of 10–20.
- More than type. The more than type uses the lower limits: 20 or more starts at the lower limit of 20–30.
- Both of them. Each type has its own limits: upper for less than, lower for more than.