Find the standard deviation and the variance of the following datasets.
| Value | Frequency |
|---|---|
| 3 | 2 |
| 5 | 1 |
| 6 | 4 |
| 9 | 3 |
The variance is the average of the squared differences between the data values and the mean. It is calculated with the following formula:
\sigma^{2}=\dfrac{\sum_{}^{}\left(x_{i}-\overline{x}\right)^{2}}{n}
Where x_{i} represent the data values, \overline{x} represents the mean, and n is the number of data values.
The standard deviation is the square root of the variance:
\sigma=\sqrt{\sigma^{2}}
First, calculate the mean of the data set:
\overline{x}=\dfrac{3\cdot2+5\cdot1+6\cdot4+9\cdot3}{10}=6.2
Second, calculate the sum of the squared differences between the data values and the mean:
\sum_{}^{}\left(x_{i}-\overline{x}\right)^{2}=2\cdot \left(3-6.2\right)^{2}+\left(5-6.2\right)^{2}+4\cdot \left(6-6.2\right)^{2}+3\cdot \left(9-6.2\right)^{2}=45.6
The number of data values is:
n=10
Therefore, the variance is:
\sigma^{2}=\dfrac{45.6}{10}=4.56
Calculate the standard deviation by square rooting the variance:
\sigma=\sqrt{4.56}\approx2.135
- \sigma\approx 2.135
- \sigma^{2}=4.56
| Value | Frequency |
|---|---|
| 4 | 2 |
| 7 | 2 |
| 8 | 1 |
| 10 | 5 |
The variance is the average of the squared differences between the data values and the mean. It is calculated with the following formula:
\sigma^{2}=\dfrac{\sum_{}^{}\left(x_{i}-\overline{x}\right)^{2}}{n}
Where x_{i} represent the data values, \overline{x} represents the mean, and n is the number of data values.
The standard deviation is the square root of the variance:
\sigma=\sqrt{\sigma^{2}}
First, calculate the mean of the data set:
\overline{x}=\dfrac{4\cdot2+7\cdot2+8\cdot1+10\cdot5}{10}=8
Second, calculate the sum of the squared differences between the data values and the mean:
\sum_{}^{}\left(x_{i}-\overline{x}\right)^{2}=2\cdot \left(4-8\right)^{2}+2\cdot\left(7-8\right)^{2}+1\cdot \left(8-8\right)^{2}+5\cdot \left(10-8\right)^{2}=54
The number of data values is:
n=10
Therefore, the variance is:
\sigma^{2}=\dfrac{54}{10}=5.4
Calculate the standard deviation by square rooting the variance:
\sigma=\sqrt{5.4}\approx2.324.
- \sigma\approx 2.324
- \sigma^{2}=5.4
| Value | Frequency |
|---|---|
| 7 | 4 |
| 8 | 5 |
| 9 | 6 |
| 10 | 5 |
The variance is the average of the squared differences between the data values and the mean. It is calculated with the following formula:
\sigma^{2}=\dfrac{\sum_{}^{}\left(x_{i}-\overline{x}\right)^{2}}{n}
Where x_{i} represent the data values, \overline{x} represents the mean, and n is the number of data values.
The standard deviation is the square root of the variance:
\sigma=\sqrt{\sigma^{2}}
First, calculate the mean of the data set:
\overline{x}=\dfrac{7\cdot4+8\cdot5+9\cdot6+10\cdot5}{20}=8.6
Second, calculate the sum of the squared differences between the data values and the mean:
\sum_{}^{}\left(x_{i}-\overline{x}\right)^{2}=4\cdot \left(7-8.6\right)^{2}+5\cdot\left(8-8.6\right)^{2}+6\cdot \left(9-8.6\right)^{2}+5\cdot \left(10-8.6\right)^{2}=22.8
The number of data values is:
n=20
Therefore, the variance is:
\sigma^{2}=\dfrac{22.8}{20}=1.14
Calculate the standard deviation by square rooting the variance:
\sigma=\sqrt{1.14}\approx1.068.
- \sigma\approx 1.068
- \sigma^{2}=1.14
| Value | Frequency |
|---|---|
| 5 | 1 |
| 8 | 2 |
| 10 | 3 |
| 12 | 4 |
The variance is the average of the squared differences between the data values and the mean. It is calculated with the following formula:
\sigma^{2}=\dfrac{\sum_{}^{}\left(x_{i}-\overline{x}\right)^{2}}{n}
Where x_{i} represent the data values, \overline{x} represents the mean, and n is the number of data values.
The standard deviation is the square root of the variance:
\sigma=\sqrt{\sigma^{2}}
First, calculate the mean of the data set:
\overline{x}=\dfrac{5\cdot1+8\cdot2+10\cdot3+12\cdot4}{10}=9.9
Second, calculate the sum of the squared differences between the data values and the mean:
\sum_{}^{}\left(x_{i}-\overline{x}\right)^{2}=1\cdot \left(5-9.9\right)^{2}+2\cdot\left(8-9.9\right)^{2}+3\cdot \left(10-9.9\right)^{2}+4\cdot \left(12-9.9\right)^{2}=48.9
The number of data values is:
n=10
Therefore, the variance is:
\sigma^{2}=\dfrac{48.9}{10}=4.89
Calculate the standard deviation by square rooting the variance:
\sigma=\sqrt{4.89}\approx2.21.
- \sigma\approx 2.21
- \sigma^{2}=4.89
| Value | Frequency |
|---|---|
| 10 | 4 |
| 20 | 6 |
| 30 | 4 |
| 40 | 2 |
The variance is the average of the squared differences between the data values and the mean. It is calculated with the following formula:
\sigma^{2}=\dfrac{\sum_{}^{}\left(x_{i}-\overline{x}\right)^{2}}{n}
Where x_{i} represent the data values, \overline{x} represents the mean, and n is the number of data values.
The standard deviation is the square root of the variance:
\sigma=\sqrt{\sigma^{2}}
First, calculate the mean of the data set:
\overline{x}=\dfrac{10\cdot4+20\cdot6+30\cdot4+40\cdot2}{16}=22.5
Second, calculate the sum of the squared differences between the data values and the mean:
\sum_{}^{}\left(x_{i}-\overline{x}\right)^{2}=4\cdot \left(10-22.5\right)^{2}+6\cdot\left(20-22.5\right)^{2}+4\cdot \left(30-22.5\right)^{2}+2\cdot \left(40-22.5\right)^{2}=1\ 500
The number of data values is:
n=16
Therefore, the variance is:
\sigma^{2}=\dfrac{1\ 500}{16}=93.75
Calculate the standard deviation by square rooting the variance:
\sigma=\sqrt{93.75}\approx9.68.
- \sigma\approx 9.68
- \sigma^{2}=93.75
| Value | Frequency |
|---|---|
| 2 | 10 |
| 1 | 20 |
| 3 | 40 |
| 4 | 30 |
The variance is the average of the squared differences between the data values and the mean. It is calculated with the following formula:
\sigma^{2}=\dfrac{\sum_{}^{}\left(x_{i}-\overline{x}\right)^{2}}{n}
Where x_{i} represent the data values, \overline{x} represents the mean, and n is the number of data values.
The standard deviation is the square root of the variance:
\sigma=\sqrt{\sigma^{2}}
First, calculate the mean of the data set:
\overline{x}=\dfrac{2\cdot10+1\cdot20+3\cdot40+4\cdot30}{100}=2.8
Second, calculate the sum of the squared differences between the data values and the mean:
\sum_{}^{}\left(x_{i}-\overline{x}\right)^{2}=10\cdot \left(2-2.8\right)^{2}+20\cdot\left(1-2.8\right)^{2}+40\cdot \left(3-2.8\right)^{2}+30\cdot \left(4-2.8\right)^{2}=116
The number of data values is:
n=100
Therefore, the variance is:
\sigma^{2}=\dfrac{116}{100}=1.16
Calculate the standard deviation by square rooting the variance:
\sigma=\sqrt{1.16}\approx1.08.
- \sigma\approx 1.08
- \sigma^{2}=1.16