The table below contains the class midpoints and the corresponding class frequencies.
(a) Complete the table below.
| Class Midpoint $x$ | Class Frequency $f$ | $f \cdot x$ | $f \cdot x^2$ |
|---|---|---|---|
| 4 | 3 | | |
| 7 | 5 | | |
| 10 | 2 | | |
Answer (2)
To complete the table, we calculated f ⋅ x and f ⋅ x 2 for each row. The completed values are 12, 35, and 20 for f ⋅ x and 48, 245, and 200 for f ⋅ x 2 . The final completed table reflects these calculations.
;
Calculate f "." x for each row: 3"."4 = 12 , 5"."7 = 35 , 2"."10 = 20 .
Calculate f "." x 2 for each row: 3"." 4 2 = 48 , 5"." 7 2 = 245 , 2"."1 0 2 = 200 .
Complete the table with the calculated values.
The completed values for f "." x and f "." x 2 are 12 , 48 , 35 , 245 , 20 , 200 .
Explanation
Understand the problem and provided data We are given a table with class midpoints x and their corresponding class frequencies f . Our task is to complete the table by calculating the values of f "." x and f "." x 2 for each row.
Calculations for the first row For the first row, we have x = 4 and f = 3 . We need to calculate f "." x and f "." x 2 .
Calculate f * x Calculating f "." x for the first row: f "." x = 3"."4 = 12
Calculate f * x^2 Calculating f "." x 2 for the first row: f "." x 2 = 3"." 4 2 = 3"."16 = 48
Calculations for the second row For the second row, we have x = 7 and f = 5 . We need to calculate f "." x and f "." x 2 .
Calculate f * x Calculating f "." x for the second row: f "." x = 5"."7 = 35
Calculate f * x^2 Calculating f "." x 2 for the second row: f "." x 2 = 5"." 7 2 = 5"."49 = 245
Calculations for the third row For the third row, we have x = 10 and f = 2 . We need to calculate f "." x and f "." x 2 .
Calculate f * x Calculating f "." x for the third row: f "." x = 2"."10 = 20
Calculate f * x^2 Calculating f "." x 2 for the third row: f "." x 2 = 2"."1 0 2 = 2"."100 = 200
Final Answer The completed table is as follows:
Class Midpoint x
Class Frequency f
f "." x
f "." x 2
4
3
12
48
7
5
35
245
10
2
20
200
Examples
This type of calculation is used in statistics to find the mean and variance of grouped data. For example, if you have survey data grouped into age ranges (like 20-30, 30-40, etc.), you can use the midpoint of each range as x and the number of respondents in that range as f . Then, calculating f "." x and f "." x 2 helps you estimate the average age and the spread of ages in the survey.
