Advertisements
Advertisements
प्रश्न
Show that the two formulae for the standard deviation of ungrouped data
\[\sigma = \sqrt{\frac{1}{n} \sum \left( x_i - X \right)^2_{}}\] and
\[\sigma' = \sqrt{\frac{1}{n} \sum x_i^2 - X^2_{}}\] are equivalent, where \[X = \frac{1}{n}\sum_{} x_i\]
Advertisements
उत्तर
\[\sigma = \sqrt{\frac{1}{n} \sum \left( x_i - X \right)^2_{}}\]
\[ = \sqrt{\frac{1}{n} \sum \left( x_i^2 - 2 x_i X + X^2 \right)_{}}\]
\[ = \sqrt{\frac{1}{n} \sum x^2 _i- \frac{1}{n}\sum 2 x_i X + \frac{1}{n}\sum X^2}\]
\[ = \sqrt{\frac{1}{n} \sum ^2 x_i - \frac{1}{n} \times 2X\sum x_i + \frac{1}{n} \times X^2 \sum 1}\]
\[ = \sqrt{\frac{1}{n} \sum x^2_i - \frac{1}{n} \times 2X \times nX + \frac{1}{n} \times X^2 \times n} \left( X = \frac{1}{n} \sum x_i \right)\]
\[= \sqrt{\frac{1}{n} \sum x_i^2 - 2 X^2 +_{} X^2}\]
\[ = \sqrt{\frac{1}{n} \sum x_i^2 - X^2_{}}\]
\[ = \sigma'\]
Hence, the formulae
\[\sigma = \sqrt{\frac{1}{n} \sum \left( x_i - X \right)^2_{}}\] and
\[\sigma' = \sqrt{\frac{1}{n} \sum x_i^2 - X^2_{}}\] are equivalent, where
\[X = \frac{1}{n}\sum_{} x_i\].
APPEARS IN
संबंधित प्रश्न
Find the mean and variance for the data.
6, 7, 10, 12, 13, 4, 8, 12
Find the mean and variance for the data.
| xi | 6 | 10 | 14 | 18 | 24 | 28 | 30 |
| fi | 2 | 4 | 7 | 12 | 8 | 4 | 3 |
Find the mean and variance for the data.
| xi | 92 | 93 | 97 | 98 | 102 | 104 | 109 |
| fi | 3 | 2 | 3 | 2 | 6 | 3 | 3 |
The following is the record of goals scored by team A in a football session:
|
No. of goals scored |
0 |
1 |
2 |
3 |
4 |
|
No. of matches |
1 |
9 |
7 |
5 |
3 |
For the team B, mean number of goals scored per match was 2 with a standard deviation 1.25 goals. Find which team may be considered more consistent?
Given that `barx` is the mean and σ2 is the variance of n observations x1, x2, …,xn. Prove that the mean and variance of the observations ax1, ax2, ax3, …,axn are `abarx` and a2 σ2, respectively (a ≠ 0).
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.
Find the mean, variance and standard deviation for the data 15, 22, 27, 11, 9, 21, 14, 9.
The variance of 20 observations is 5. If each observation is multiplied by 2, find the variance of the resulting observations.
The mean and variance of 8 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 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?
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:
(i) If wrong item is omitted
(ii) if it is replaced by 12.
Find the standard deviation for the following distribution:
| x : | 4.5 | 14.5 | 24.5 | 34.5 | 44.5 | 54.5 | 64.5 |
| f : | 1 | 5 | 12 | 22 | 17 | 9 | 4 |
A student obtained the mean and standard deviation of 100 observations as 40 and 5.1 respectively. It was later found that one observation was wrongly copied as 50, the correct figure being 40. Find the correct mean and S.D.
The means and standard deviations of heights ans weights of 50 students of a class are as follows:
| Weights | Heights | |
| Mean | 63.2 kg | 63.2 inch |
| Standard deviation | 5.6 kg | 11.5 inch |
Which shows more variability, heights or weights?
Coefficient of variation of two distributions are 60% and 70% and their standard deviations are 21 and 16 respectively. What are their arithmetic means?
From the data given below state which group is more variable, G1 or G2?
| Marks | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |
| Group G1 | 9 | 17 | 32 | 33 | 40 | 10 | 9 |
| Group G2 | 10 | 20 | 30 | 25 | 43 | 15 | 7 |
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 the sum of the squares of deviations for 10 observations taken from their mean is 2.5, then write the value of standard deviation.
In a series of 20 observations, 10 observations are each equal to k and each of the remaining half is equal to − k. If the standard deviation of the observations is 2, then write the value of k.
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
The standard deviation of first 10 natural numbers is
The mean of 100 observations is 50 and their standard deviation is 5. The sum of all squares of all the observations 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.
Find the standard deviation of the first n natural numbers.
The mean and standard deviation of some data for the time taken to complete a test are calculated with the following results:
Number of observations = 25, mean = 18.2 seconds, standard deviation = 3.25 seconds. Further, another set of 15 observations x1, x2, ..., x15, also in seconds, is now available and we have `sum_(i = 1)^15 x_i` = 279 and `sum_(i = 1)^15 x^2` = 5524. Calculate the standard derivation based on all 40 observations.
The mean life of a sample of 60 bulbs was 650 hours and the standard deviation was 8 hours. A second sample of 80 bulbs has a mean life of 660 hours and standard deviation 7 hours. Find the overall standard deviation.
If for distribution `sum(x - 5)` = 3, `sum(x - 5)^2` = 43 and total number of items is 18. Find the mean and standard deviation.
Let x1, x2, ..., xn be n observations and `barx` be their arithmetic mean. The formula for the standard deviation is given by ______.
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 ______.
Standard deviations for first 10 natural numbers is ______.
Coefficient of variation of two distributions are 50 and 60, and their arithmetic means are 30 and 25 respectively. Difference of their standard deviation is ______.
If the variance of a data is 121, then the standard deviation of the data is ______.
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.
