Find the mean deviation about the mean of the distribution:
| Size | 20 | 21 | 22 | 23 | 24 |
|---|---|---|---|---|---|
| Frequency | 6 | 4 | 5 | 1 | 4 |
0.32
Find the mean deviation about the median of the following distribution:
| Marks obtained | 10 | 11 | 12 | 14 | 15 |
|---|---|---|---|---|---|
| No. of students | 2 | 3 | 8 | 3 | 4 |
1.25
Calculate the mean deviation about the mean of the set of first \( n \) natural numbers when \( n \) is an odd number.
\( \dfrac{n^2 - 1}{4n} \)
Calculate the mean deviation about the mean of the set of first \( n \) natural numbers when \( n \) is an even number.
\( \dfrac{n}{4} \)
Find the standard deviation of the first \( n \) natural numbers.
\( \sqrt{\dfrac{n^2 - 1}{12}} \)
The mean and standard deviation of 25 observations are 18.2 seconds and 3.25 seconds. A second set of 15 observations has \( \sum x_i = 279 \) and \( \sum x_i^2 = 5524 \). Calculate the standard deviation of all 40 observations.
3.87
The mean and standard deviation of a set of \( n_1 \) observations are \( \bar{x}_1 \) and \( s_1 \). For another set of \( n_2 \) observations, the mean and standard deviation are \( \bar{x}_2 \) and \( s_2 \). Show that the standard deviation of the combined set is:
\( \sqrt{\dfrac{n_1(s_1)^2 + n_2(s_2)^2}{n_1 + n_2} + \dfrac{n_1 n_2 (\bar{x}_1 - \bar{x}_2)^2}{(n_1 + n_2)^2}} \)
\( \sqrt{\dfrac{n_1(s_1)^2 + n_2(s_2)^2}{n_1 + n_2} + \dfrac{n_1 n_2 (\bar{x}_1 - \bar{x}_2)^2}{(n_1 + n_2)^2}} \)
Two sets each of 20 observations have the same standard deviation 5. The first set has a mean 17 and the second a mean 22. Determine the standard deviation of the combined set.
5.59
The frequency distribution:
| x | A | 2A | 3A | 4A | 5A | 6A |
|---|---|---|---|---|---|---|
| f | 2 | 1 | 1 | 1 | 1 | 1 |
has variance 160. Determine the value of \( A \).
7
For the frequency distribution:
| x | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|
| f | 4 | 9 | 16 | 14 | 11 | 6 |
Find the standard deviation.
1.38
The following is the frequency distribution of marks in a class of 60 students:
| Marks | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Frequency | x - 2 | x | x^2 | (x + 1)^2 | 2x | x + 1 |
Determine the mean and standard deviation where \( x \) is a positive integer.
Mean = 2.8, SD = 1.12
The mean life of a sample of 60 bulbs is 650 hours with standard deviation 8 hours. A second sample of 80 bulbs has mean life 660 hours with standard deviation 7 hours. Find the overall standard deviation.
8.9
A set of 100 items has mean 50 and standard deviation 4. Find the sum of all items and the sum of squares of all items.
5000, 251600
For a distribution where \( \sum (x - 5) = 3 \) and \( \sum (x - 5)^2 = 43 \), and the total number of items is 18, find the mean and standard deviation.
Mean = 5.17, SD = 1.53
Find the mean and variance of the following frequency distribution:
| x | 1 ≤ x < 3 | 3 ≤ x < 5 | 5 ≤ x < 7 | 7 ≤ x < 10 |
|---|---|---|---|---|
| f | 6 | 4 | 5 | 1 |
Mean = 5.5, Var = 4.26