"Sample standard deviation"

Topic Overview

• This section discusses how to compute the sample standard deviation in statistics.

Sample Standard Deviation

• The sample standard deviation, denoted as $s$, measures the spread of sample measurements around their mean, denoted as $\bar{x}$.

Steps to Compute Sample Standard Deviation

1. Calculate the Mean (Average):

   • To find the mean of a sample, sum all the values and divide by the number of values in the sample.
   • Example: For the responses $8, 6, 9, 4, 8$, compute the mean as follows:
     $\bar{x} = \frac{8 + 6 + 9 + 4 + 8}{5} = 7$

2. Find the Differences from the Mean:

   • Subtract the mean from each sample value:
     • $8 - 7 = 1$
     • $6 - 7 = -1$
     • $9 - 7 = 2$
     • $4 - 7 = -3$
     • $8 - 7 = 1$

3. Square the Differences:

   • Square each of the differences obtained in the previous step:
     • $(1)^2 = 1$
     • $(-1)^2 = 1$
     • $(2)^2 = 4$
     • $(-3)^2 = 9$
     • $(1)^2 = 1$

4. Sum the Squared Differences:

   • Combine all squared differences:
     $1 + 1 + 4 + 9 + 1 = 16$

5. Calculate the Variance:

   • The formula for sample variance is:
     $s^2 = \frac{\sum<em>{i=1}^{n} (x</em>i - \bar{x})^2}{n - 1}$
   • Here, $n$ is the number of observations in the sample (5 in this case), hence:
     $s^2 = \frac{16}{5 - 1} = 4$

6. Take the Square Root for Standard Deviation:

   • Finally, the sample standard deviation is:
     $s = \sqrt{s^2} = \sqrt{4} = 2$

Result

• The computed sample standard deviation is:
  $s = 2.00$

General Formula for Sample Standard Deviation

• The general formula for the sample standard deviation can be expressed as:
  $s = \sqrt{\frac{\sum<em>{i=1}^{n} (x</em>i - \bar{x})^2}{n - 1}}$

• Where:

  • $x_i$ are the individual sample points
  • $\bar{x}$ is the mean of the sample
  • $n$ is the number of sample points