Standard Deviation worksheet

1. Population standard deviation (σ): 2.22

1. Find the mean (the average, x̄): add up the 8 values and divide by how many there are, n

   $\mathrm{x̄}$$=\frac{3+4+3+3+5+4+6+10}{8}$$=4.75$

2. See how far each value is from the mean. Square each gap so minus signs don't cancel, then add them up (SS)

   $\mathrm{SS}$$={\left(3-4.75\right)}^2+{\left(4-4.75\right)}^2+\text{…}+{\left(10-4.75\right)}^2$$=3.0625+0.5625+3.0625+3.0625+0.0625+0.5625+1.5625+27.5625$$=39.5$

3. Divide SS by n to get the population variance, the average squared gap

   ${\sigma }^2$$=\frac{39.5}{8}$$=4.9375$

4. Take the square root of the variance. This undoes the squaring and puts the answer back in your data's units

   $\sigma$$=\sqrt{4.9375}$$=2.222049$

5. Extra: divide the sample standard deviation by the square root of n to get the standard error (SE) of the mean

   $\mathrm{SE}$$=\frac{2.37547}{\sqrt{8}}$$=0.839855$

2. Population standard deviation (σ): 2

1. Find the mean (the average, x̄): add up the 8 values and divide by how many there are, n

   $\mathrm{x̄}$$=\frac{2+4+4+4+5+5+7+9}{8}$$=5$

2. See how far each value is from the mean. Square each gap so minus signs don't cancel, then add them up (SS)

   $\mathrm{SS}$$={\left(2-5\right)}^2+{\left(4-5\right)}^2+\text{…}+{\left(9-5\right)}^2$$=9+1+1+1+0+0+4+16$$=32$

3. Divide SS by n to get the population variance, the average squared gap

   ${\sigma }^2$$=\frac{32}{8}$$=4$

4. Take the square root of the variance. This undoes the squaring and puts the answer back in your data's units

   $\sigma$$=\sqrt{4}$$=2$

5. Extra: divide the sample standard deviation by the square root of n to get the standard error (SE) of the mean

   $\mathrm{SE}$$=\frac{2.13809}{\sqrt{8}}$$=0.755929$

3. Sample standard deviation (s): 2.14

1. Find the mean (the average, x̄): add up the 8 values and divide by how many there are, n

   $\mathrm{x̄}$$=\frac{2+4+4+4+5+5+7+9}{8}$$=5$

2. See how far each value is from the mean. Square each gap so minus signs don't cancel, then add them up (SS)

   $\mathrm{SS}$$={\left(2-5\right)}^2+{\left(4-5\right)}^2+\text{…}+{\left(9-5\right)}^2$$=9+1+1+1+0+0+4+16$$=32$

3. Divide SS by n − 1 to get the sample variance. Using one less than n corrects for working from a sample

   $s^2$$=\frac{32}{8\text{− 1}}$$=\frac{32}{7}$$=4.571429$

4. Take the square root of the variance. This undoes the squaring and puts the answer back in your data's units

   $s$$=\sqrt{4.571429}$$=2.13809$

5. Extra: divide the sample standard deviation by the square root of n to get the standard error (SE) of the mean

   $\mathrm{SE}$$=\frac{2.13809}{\sqrt{8}}$$=0.755929$

4. Sample standard deviation (s): 2.88

1. Find the mean (the average, x̄): add up the 8 values and divide by how many there are, n

   $\mathrm{x̄}$$=\frac{2+3+4+4+7+6+9+10}{8}$$=5.625$

2. See how far each value is from the mean. Square each gap so minus signs don't cancel, then add them up (SS)

   $\mathrm{SS}$$={\left(2-5.625\right)}^2+{\left(3-5.625\right)}^2+\text{…}+{\left(10-5.625\right)}^2$$=13.140625+6.890625+2.640625+2.640625+1.890625+0.140625+11.390625+19.140625$$=57.875$

3. Divide SS by n − 1 to get the sample variance. Using one less than n corrects for working from a sample

   $s^2$$=\frac{57.875}{8\text{− 1}}$$=\frac{57.875}{7}$$=8.267857$

4. Take the square root of the variance. This undoes the squaring and puts the answer back in your data's units

   $s$$=\sqrt{8.267857}$$=2.875388$

5. Extra: divide the sample standard deviation by the square root of n to get the standard error (SE) of the mean

   $\mathrm{SE}$$=\frac{2.875388}{\sqrt{8}}$$=1.016603$

5. Population standard deviation (σ): 1.93

1. Find the mean (the average, x̄): add up the 8 values and divide by how many there are, n

   $\mathrm{x̄}$$=\frac{2+4+4+3+4+3+7+8}{8}$$=4.375$

2. See how far each value is from the mean. Square each gap so minus signs don't cancel, then add them up (SS)

   $\mathrm{SS}$$={\left(2-4.375\right)}^2+{\left(4-4.375\right)}^2+\text{…}+{\left(8-4.375\right)}^2$$=5.640625+0.140625+0.140625+1.890625+0.140625+1.890625+6.890625+13.140625$$=29.875$

3. Divide SS by n to get the population variance, the average squared gap

   ${\sigma }^2$$=\frac{29.875}{8}$$=3.734375$

4. Take the square root of the variance. This undoes the squaring and puts the answer back in your data's units

   $\sigma$$=\sqrt{3.734375}$$=1.932453$

5. Extra: divide the sample standard deviation by the square root of n to get the standard error (SE) of the mean

   $\mathrm{SE}$$=\frac{2.065879}{\sqrt{8}}$$=0.730399$

6. Sample standard deviation (s): 2.12

1. Find the mean (the average, x̄): add up the 8 values and divide by how many there are, n

   $\mathrm{x̄}$$=\frac{2+4+4+3+6+5+5+9}{8}$$=4.75$

2. See how far each value is from the mean. Square each gap so minus signs don't cancel, then add them up (SS)

   $\mathrm{SS}$$={\left(2-4.75\right)}^2+{\left(4-4.75\right)}^2+\text{…}+{\left(9-4.75\right)}^2$$=7.5625+0.5625+0.5625+3.0625+1.5625+0.0625+0.0625+18.0625$$=31.5$

3. Divide SS by n − 1 to get the sample variance. Using one less than n corrects for working from a sample

   $s^2$$=\frac{31.5}{8\text{− 1}}$$=\frac{31.5}{7}$$=4.5$

4. Take the square root of the variance. This undoes the squaring and puts the answer back in your data's units

   $s$$=\sqrt{4.5}$$=2.12132$

5. Extra: divide the sample standard deviation by the square root of n to get the standard error (SE) of the mean

   $\mathrm{SE}$$=\frac{2.12132}{\sqrt{8}}$$=0.75$

7. Sample standard deviation (s): 1.77

1. Find the mean (the average, x̄): add up the 8 values and divide by how many there are, n

   $\mathrm{x̄}$$=\frac{2+4+4+3+4+5+7+7}{8}$$=4.5$

2. See how far each value is from the mean. Square each gap so minus signs don't cancel, then add them up (SS)

   $\mathrm{SS}$$={\left(2-4.5\right)}^2+{\left(4-4.5\right)}^2+\text{…}+{\left(7-4.5\right)}^2$$=6.25+0.25+0.25+2.25+0.25+0.25+6.25+6.25$$=22$

3. Divide SS by n − 1 to get the sample variance. Using one less than n corrects for working from a sample

   $s^2$$=\frac{22}{8\text{− 1}}$$=\frac{22}{7}$$=3.142857$

4. Take the square root of the variance. This undoes the squaring and puts the answer back in your data's units

   $s$$=\sqrt{3.142857}$$=1.772811$

5. Extra: divide the sample standard deviation by the square root of n to get the standard error (SE) of the mean

   $\mathrm{SE}$$=\frac{1.772811}{\sqrt{8}}$$=0.626783$

8. Sample standard deviation (s): 1.96

1. Find the mean (the average, x̄): add up the 8 values and divide by how many there are, n

   $\mathrm{x̄}$$=\frac{2+3+4+5+5+5+8+7}{8}$$=4.875$

2. See how far each value is from the mean. Square each gap so minus signs don't cancel, then add them up (SS)

   $\mathrm{SS}$$={\left(2-4.875\right)}^2+{\left(3-4.875\right)}^2+\text{…}+{\left(7-4.875\right)}^2$$=8.265625+3.515625+0.765625+0.015625+0.015625+0.015625+9.765625+4.515625$$=26.875$

3. Divide SS by n − 1 to get the sample variance. Using one less than n corrects for working from a sample

   $s^2$$=\frac{26.875}{8\text{− 1}}$$=\frac{26.875}{7}$$=3.839286$

4. Take the square root of the variance. This undoes the squaring and puts the answer back in your data's units

   $s$$=\sqrt{3.839286}$$=1.95941$

5. Extra: divide the sample standard deviation by the square root of n to get the standard error (SE) of the mean

   $\mathrm{SE}$$=\frac{1.95941}{\sqrt{8}}$$=0.692756$

9. Sample standard deviation (s): 2.27

1. Find the mean (the average, x̄): add up the 8 values and divide by how many there are, n

   $\mathrm{x̄}$$=\frac{1+4+5+5+5+4+8+8}{8}$$=5$

2. See how far each value is from the mean. Square each gap so minus signs don't cancel, then add them up (SS)

   $\mathrm{SS}$$={\left(1-5\right)}^2+{\left(4-5\right)}^2+\text{…}+{\left(8-5\right)}^2$$=16+1+0+0+0+1+9+9$$=36$

3. Divide SS by n − 1 to get the sample variance. Using one less than n corrects for working from a sample

   $s^2$$=\frac{36}{8\text{− 1}}$$=\frac{36}{7}$$=5.142857$

4. Take the square root of the variance. This undoes the squaring and puts the answer back in your data's units

   $s$$=\sqrt{5.142857}$$=2.267787$

5. Extra: divide the sample standard deviation by the square root of n to get the standard error (SE) of the mean

   $\mathrm{SE}$$=\frac{2.267787}{\sqrt{8}}$$=0.801784$

10. Population standard deviation (σ): 2.44

1. Find the mean (the average, x̄): add up the 8 values and divide by how many there are, n

   $\mathrm{x̄}$$=\frac{2+4+5+4+5+5+6+11}{8}$$=5.25$

2. See how far each value is from the mean. Square each gap so minus signs don't cancel, then add them up (SS)

   $\mathrm{SS}$$={\left(2-5.25\right)}^2+{\left(4-5.25\right)}^2+\text{…}+{\left(11-5.25\right)}^2$$=10.5625+1.5625+0.0625+1.5625+0.0625+0.0625+0.5625+33.0625$$=47.5$

3. Divide SS by n to get the population variance, the average squared gap

   ${\sigma }^2$$=\frac{47.5}{8}$$=5.9375$

4. Take the square root of the variance. This undoes the squaring and puts the answer back in your data's units

   $\sigma$$=\sqrt{5.9375}$$=2.436699$

5. Extra: divide the sample standard deviation by the square root of n to get the standard error (SE) of the mean

   $\mathrm{SE}$$=\frac{2.60494}{\sqrt{8}}$$=0.920985$