Advertisements
Advertisements
प्रश्न
Find the standard deviation of the first n natural numbers.
Advertisements
उत्तर
| `x_i` | 1 | 2 | 3 | 4 | 5 | – | – | n |
| `x_i^2` | 1 | 4 | 9 | 16 | 25 | – | – | n2 |
`sumx_i = 1 + 2 + 3 + 4 + 5 + ... + n = (n(n + 1))/2`
`sumx_i^2 = 1^2 + 2^2 + 3^2 + ... + n^2 = (n(n + 1)(2n + 1))/6`
∴ S.D. `(sigma) = sqrt((sumx_i^2)/n - ((sumx_i)/n)^2`
= `sqrt((n(n + 1)(2n + 1))/(6n) - (n^2(n + 1)^2)/(4n^2))`
= `sqrt(((n + 1)(2n + 1))/6 - (n + 1)^2/4`
= `sqrt((2n^2 + 3n + 1)/6 - (n^2 + 2n + 1)/4)`
= `sqrt((4n^2 + 6n + 2 - 3n^2 - 6n - 3)/12`
= `sqrt((n^2 - 1)/12)`
Hence, the required S.D. = `sqrt((n^2 - 1)/12`
APPEARS IN
संबंधित प्रश्न
Find the mean and variance for the first n natural numbers.
The sum and sum of squares corresponding to length x (in cm) and weight y (in gm) of 50 plant products are given below:
`sum_(i-1)^50 x_i = 212, sum_(i=1)^50 x_i^2 = 902.8, sum_(i=1)^50 y_i = 261, sum_(i = 1)^50 y_i^2 = 1457.6`
Which is more varying, the length or weight?
The mean and variance of eight observations are 9 and 9.25, respectively. If six of the observations are 6, 7, 10, 12, 12 and 13, find the remaining two observations.
The mean and standard deviation of six observations are 8 and 4, respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations
The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation 8 was incorrect. Calculate the correct mean and standard deviation in each of the following cases:
- If wrong item is omitted.
- If it is replaced by 12.
The mean and standard deviation of a group of 100 observations were found to be 20 and 3, respectively. Later on it was found that three observations were incorrect, which were recorded as 21, 21 and 18. Find the mean and standard deviation if the incorrect observations are omitted.
Find the mean, variance and standard deviation for the data:
2, 4, 5, 6, 8, 17.
Find the mean, variance and standard deviation for the data:
6, 7, 10, 12, 13, 4, 8, 12.
Find the mean, variance and standard deviation for the data:
227, 235, 255, 269, 292, 299, 312, 321, 333, 348.
The variance of 20 observations is 5. If each observation is multiplied by 2, find the variance of the resulting observations.
The mean and standard deviation of 100 observations were calculated as 40 and 5.1 respectively by a student who took by mistake 50 instead of 40 for one observation. What are the correct mean and standard deviation?
Find the standard deviation for the following data:
| x : | 3 | 8 | 13 | 18 | 23 |
| f : | 7 | 10 | 15 | 10 | 6 |
Calculate the mean and S.D. for the following data:
| Expenditure in Rs: | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
| Frequency: | 14 | 13 | 27 | 21 | 15 |
Calculate the standard deviation for the following data:
| Class: | 0-30 | 30-60 | 60-90 | 90-120 | 120-150 | 150-180 | 180-210 |
| Frequency: | 9 | 17 | 43 | 82 | 81 | 44 | 24 |
Calculate the A.M. and S.D. for the following distribution:
| Class: | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |
| Frequency: | 18 | 16 | 15 | 12 | 10 | 5 | 2 | 1 |
Two plants A and B of a factory show following results about the number of workers and the wages paid to them
| Plant A | Plant B | |
| No. of workers | 5000 | 6000 |
| Average monthly wages | Rs 2500 | Rs 2500 |
| Variance of distribution of wages | 81 | 100 |
In which plant A or B is there greater variability in individual wages?
The mean and standard deviation of marks obtained by 50 students of a class in three subjects, mathematics, physics and chemistry are given below:
| Subject | Mathematics | Physics | Chemistry |
| Mean | 42 | 32 | 40.9 |
| Standard Deviation | 12 | 15 | 20 |
Which of the three subjects shows the highest variability in marks and which shows the lowest?
Find the coefficient of variation for the following data:
| Size (in cms): | 10-15 | 15-20 | 20-25 | 25-30 | 30-35 | 35-40 |
| No. of items: | 2 | 8 | 20 | 35 | 20 | 15 |
If each observation of a raw data whose standard deviation is σ is multiplied by a, then write the S.D. of the new set of observations.
If v is the variance and σ is the standard deviation, then
If the standard deviation of a variable X is σ, then the standard deviation of variable \[\frac{a X + b}{c}\] is
If the S.D. of a set of observations is 8 and if each observation is divided by −2, the S.D. of the new set of observations will be
Let a, b, c, d, e be the 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
Let x1, x2, ..., xn be n observations. Let \[y_i = a x_i + b\] for i = 1, 2, 3, ..., n, where a and b are constants. If the mean of \[x_i 's\] is 48 and their standard deviation is 12, the mean of \[y_i 's\] is 55 and standard deviation of \[y_i 's\] is 15, the values of a and b are
The standard deviation of the observations 6, 5, 9, 13, 12, 8, 10 is
Show that the two formulae for the standard deviation of ungrouped data.
`sigma = sqrt((x_i - barx)^2/n)` and `sigma`' = `sqrt((x^2_i)/n - barx^2)` are equivalent.
The standard deviation of the data 6, 5, 9, 13, 12, 8, 10 is ______.
Let x1, x2, ... xn be n observations. Let wi = lxi + k for i = 1, 2, ...n, where l and k are constants. If the mean of xi’s is 48 and their standard deviation is 12, the mean of wi’s is 55 and standard deviation of wi’s is 15, the values of l and k should be ______.
The standard deviation of a data is ______ of any change in orgin, but is ______ on the change of scale.
The standard deviation is ______to the mean deviation taken from the arithmetic mean.
The mean and standard deviation of six observations are 8 and 4, respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations.
