"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 ar{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:
     ar{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{i=1}^{n} (xi - \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{i=1}^{n} (xi - \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