Standard deviation for the following distribution is:Size of Item 6 7 8 9 10 11 12 Frequency 3 6 9 13 8 5 4
1.6
Standard Deviation Calculation for Frequency Distribution
This solution explains how to find the standard deviation for a given frequency distribution. We will break down the process step-by-step, calculating the mean, variance, and finally the standard deviation.
Distribution Data Overview
We are given the 'Size of Item' (denoted as x) and the corresponding 'Frequency' (denoted as f) for different items. This data represents how often each item size occurs in a dataset.
| Size of Item (x) | Frequency (f) |
| 6 | 3 |
| 7 | 6 |
| 8 | 9 |
| 9 | 13 |
| 10 | 8 |
| 11 | 5 |
| 12 | 4 |
Calculate the Mean ($\bar{x}$)
To calculate the standard deviation, we first need to find the mean ($\bar{x}$) of the distribution. The mean is calculated by summing the product of each item size and its frequency ($\sum fx$) and then dividing by the total number of observations ($N$, which is the sum of all frequencies, $\sum f$).
The formula for the mean is:
$$ \bar{x} = \frac{\sum fx}{N} $$
Let's organize the calculations in a table:
| Size of Item (x) | Frequency (f) | fx | x - $\bar{x}$ | (x - $\bar{x}$)$^2$ | f(x - $\bar{x}$)$^2$ |
| 6 | 3 | $6 \times 3 = 18$ | $6 - 9 = -3$ | $(-3)^2 = 9$ | $3 \times 9 = 27$ |
| 7 | 6 | $7 \times 6 = 42$ | $7 - 9 = -2$ | $(-2)^2 = 4$ | $6 \times 4 = 24$ |
| 8 | 9 | $8 \times 9 = 72$ | $8 - 9 = -1$ | $(-1)^2 = 1$ | $9 \times 1 = 9$ |
| 9 | 13 | $9 \times 13 = 117$ | $9 - 9 = 0$ | $(0)^2 = 0$ | $13 \times 0 = 0$ |
| 10 | 8 | $10 \times 8 = 80$ | $10 - 9 = 1$ | $(1)^2 = 1$ | $8 \times 1 = 8$ |
| 11 | 5 | $11 \times 5 = 55$ | $11 - 9 = 2$ | $(2)^2 = 4$ | $5 \times 4 = 20$ |
| 12 | 4 | $12 \times 4 = 48$ | $12 - 9 = 3$ | $(3)^2 = 9$ | $4 \times 9 = 36$ |
| Total | $N = 48$ | $\sum fx = 432$ | $\sum f(x - \bar{x})^2 = 124$ |
Using the totals from the table:
First, calculate the total frequency ($N$):
$$ N = 3 + 6 + 9 + 13 + 8 + 5 + 4 = 48 $$
Next, calculate the sum of $fx$:
$$ \sum fx = 18 + 42 + 72 + 117 + 80 + 55 + 48 = 432 $$
Now, we can find the mean:
$$ \bar{x} = \frac{432}{48} = 9 $$
Calculate the Variance ($\sigma^2$)
The variance ($\sigma^2$) measures the average squared difference from the mean. The formula for variance in a frequency distribution is:
$$ \sigma^2 = \frac{\sum f(x - \bar{x})^2}{N} $$
We use the value $\sum f(x - \bar{x})^2 = 124$ and $N = 48$ from our table:
$$ \sigma^2 = \frac{124}{48} $$
Calculating the value:
$$ \sigma^2 \approx 2.5833 $$
Calculate the Standard Deviation ($\sigma$)
The standard deviation ($\sigma$) is the square root of the variance. It provides a measure of the spread or dispersion of data points around the mean, expressed in the same units as the data.
The formula is:
$$ \sigma = \sqrt{\sigma^2} $$
Substitute the calculated variance:
$$ \sigma = \sqrt{\frac{124}{48}} $$
$$ \sigma \approx \sqrt{2.5833} $$
Calculating the square root:
$$ \sigma \approx 1.607 $$
Rounding to one decimal place, the standard deviation is approximately 1.6.
Important Questions from Variance and Standard Deviation
The sum of deviations of n numbers from 10 and 20 are p and q respectively. If (p - q)2 = 10000, then what is the value of n?
The mean and the variance of 10 observations are given to be 4 and 2 respectively. If every observation is multiplied by 2, the mean and the variance of the new series will be respectively.
If the data are moderately non-symmetrical, then which one of the following empirical relationships is correct?
If the total number of observations is 20, ∑ x i= 1000 and \(\sum {\rm{x}}_{\rm{i}}^2 = 84000\) , then what is the variance of the distribution?
Among these options, which one is NOT an example of relative measure of dispersion?
