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
Calculate the mean deviation about the mean for the following frequency distribution:
| Class interval | 0–4 | 4–8 | 8–12 | 12–16 | 16–20 |
|---|---|---|---|---|---|
| Frequency | 4 | 6 | 8 | 5 | 2 |
0.99
Calculate the mean deviation from the median of the following data:
| Class interval | 0–6 | 6–12 | 12–18 | 18–24 | 24–30 |
|---|---|---|---|---|---|
| Frequency | 4 | 5 | 3 | 6 | 2 |
7.08
Determine the mean and standard deviation for the following distribution:
| Marks | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Frequency | 1 | 6 | 6 | 8 | 8 | 2 | 2 | 3 | 0 | 2 | 1 | 0 | 0 | 0 | 1 |
Mean = \(\tfrac{239}{40}\), SD = 2.85
The weights of coffee in 70 jars are shown in the following table:
| Weight (in grams) | 200–201 | 201–202 | 202–203 | 203–204 | 204–205 | 205–206 |
|---|---|---|---|---|---|---|
| Frequency | 13 | 27 | 18 | 10 | 1 | 1 |
Determine variance and standard deviation of the above distribution.
Var. = 1.16 gm, S.D = 1.08 gm
Determine mean and standard deviation of first \( n \) terms of an A.P. whose first term is \( a \) and common difference is \( d \).
Mean = \( a + \dfrac{d(n-1)}{2} \)
Following are the marks obtained, out of 100, by two students Ravi and Hashina in 10 tests.
Ravi: 25, 50, 45, 30, 70, 42, 36, 48, 35, 60
Hashina: 10, 70, 50, 20, 95, 55, 42, 60, 48, 80
Who is more intelligent and who is more consistent?
Hashina is more intelligent and consistent
Mean and standard deviation of 100 observations were found to be 40 and 10, respectively. If at the time of calculation two observations were wrongly taken as 30 and 70 in place of 3 and 27 respectively, find the correct standard deviation.
10.24
While calculating the mean and variance of 10 readings, a student wrongly used the reading 52 for the correct reading 25. He obtained the mean and variance as 45 and 16 respectively. Find the correct mean and the variance.
Mean = 42.3, Var. = 43.81
The mean deviation of the data 3, 10, 10, 4, 7, 10, 5 from the mean is
2
2.57
3
3.75
Mean deviation for \( n \) observations \( x_1, x_2, ..., x_n \) from their mean \( \bar{x} \) is given by
\( \sum (x_i - \bar{x}) \)
\( \dfrac{1}{n} \sum |x_i - \bar{x}| \)
\( \sum (x_i - \bar{x})^2 \)
\( \dfrac{1}{n} \sum (x_i - \bar{x})^2 \)
The lives (in hours) of 5 bulbs were noted as: 1357, 1090, 1666, 1494, 1623. The mean deviation (in hours) from their mean is
178
179
220
356
Following are the marks obtained by 9 students in a mathematics test: 50, 69, 20, 33, 53, 39, 40, 65, 59. The mean deviation from the median is
9
10.5
12.67
14.76
The standard deviation of the data 6, 5, 9, 13, 12, 8, 10 is
\( \sqrt{\dfrac{52}{7}} \)
\( \dfrac{52}{7} \)
\( \sqrt{6} \)
6
Let \( x_1, x_2, ..., x_n \) be \( n \) observations and \( \bar{x} \) be their arithmetic mean. The formula for the standard deviation is
\( \sum (x_i - \bar{x})^2 \)
\( \dfrac{\sum (x_i - \bar{x})^2}{n} \)
\( \sqrt{\dfrac{\sum (x_i - \bar{x})^2}{n}} \)
\( \sqrt{\dfrac{\sum x_i^2}{n} + \bar{x}^2} \)
The mean of 100 observations is 50 and the standard deviation is 5. The sum of all squares of all the observations is
50000
250000
252500
255000
Let \( a, b, c, d, e \) be observations with mean \( m \) and standard deviation \( s \). The standard deviation of the observations \( a+k, b+k, c+k, d+k, e+k \) is
s
ks
s+k
\( \dfrac{s}{k} \)
Let \( x_1, x_2, x_3, x_4, x_5 \) be observations with mean \( m \) and standard deviation \( s \). The standard deviation of the observations \( kx_1, kx_2, kx_3, kx_4, kx_5 \) is
k + s
\( \dfrac{s}{k} \)
ks
s
Let \( x_1, x_2, ..., x_n \) be observations. Let \( w_i = lx_i + k \). If the mean of \( x_i \) is 48 and SD = 12, and the mean of \( w_i \) is 55 and SD = 15, values of \( l \) and \( k \) are
\( l = 1.25, k = -5 \)
\( l = -1.25, k = 5 \)
\( l = 2.5, k = -5 \)
\( l = 2.5, k = 5 \)
Standard deviation for the first 10 natural numbers is
5.5
3.87
2.97
2.87
Consider the numbers 1 to 10. If 1 is added to each number, the variance of the numbers so obtained is
6.5
2.87
3.87
8.25
Consider the first 10 positive integers. If each number is multiplied by \(-1\) and then 1 is added, the variance becomes
8.25
6.5
3.87
2.87
A sample of size 60 has \( \sum x = 960 \) and \( \sum x^2 = 18000 \). The variance is
6.63
16
22
44
Coefficient of variation of two distributions are 50 and 60, and their arithmetic means are 30 and 25 respectively. Difference of their standard deviations is
0
1
1.5
2.5
The standard deviation of some temperature data in °C is 5. If the data were converted into °F, the variance would be
81
57
36
25
Coefficient of variation = ____ × 100 / Mean
SD
If \( \bar{x} \) is the mean of n values of x, then \( \sum (x_i - \bar{x}) \) is always equal to ____.
If a has any value other than \( \bar{x} \), then \( \sum (x_i - \bar{x})^2 \) is ____ than \( \sum (x_i - a)^2 \).
0
less
If the variance of a data is 121, then the standard deviation of the data is ____.
11
The standard deviation of a data is ____ of any change in origin, but is ____ on the change of scale.
Independent
The sum of the squares of the deviations of the values of the variable is ____ when taken about their arithmetic mean.
Minimum
The mean deviation of the data is ____ when measured from the median.
Least
The standard deviation is ____ to the mean deviation taken from the arithmetic mean.
greater than or equal