<< Chapter < Page Chapter >> Page >

In these formulas, f represents the frequency with which a value appears. For example, if a value appears once, f is one. If a value appears three times in the data set or population, f is three.

Sampling variability of a statistic

The statistic of a sampling distribution was discussed in Descriptive Statistics: Measuring the Center of the Data . How much the statistic varies from one sample to another is known as the sampling variability of a statistic . You typically measure the sampling variability of a statistic by its standard error. The standard error of the mean is an example of a standard error. It is a special standard deviation and is known as the standard deviation of the sampling distribution of the mean. You will cover the standard error of the mean in the chapter The Central Limit Theorem (not now). The notation for the standard error of the mean is σ n where σ is the standard deviation of the population and n is the size of the sample.

Note

In practice, USE A CALCULATOR OR COMPUTER SOFTWARE TO CALCULATE THE STANDARD DEVIATION. We will concentrate on using and interpreting the information that the standard deviation gives us. However you should study the following step-by-step example to help you understand how the standard deviation measures variation from the mean.

In a fifth grade class, the teacher was interested in the average age and the sample standard deviation of the ages of her students. The following data are the ages for a SAMPLE of n = 20 fifth grade students. The ages are rounded to the nearest half year:

9; 9.5; 9.5; 10; 10; 10; 10; 10.5; 10.5; 10.5; 10.5; 11; 11; 11; 11; 11; 11; 11.5; 11.5; 11.5;

x ¯ = 9 + 9 .5(2) + 10(4) + 10 .5(4) + 11(6) + 11 .5(3) 20 = 10.525

The average age is 10.53 years, rounded to two places.

The variance may be calculated by using a table. Then the standard deviation is calculated by taking the square root of the variance. We will explain the parts of the table after calculating s .

Data Freq. Deviations Deviations 2 (Freq.)( Deviations 2 )
x f ( x x ¯ ) ( x x ¯ ) 2 ( f )( x x ¯ ) 2
9 1 9 – 10.525 = –1.525 (–1.525) 2 = 2.325625 1 × 2.325625 = 2.325625
9.5 2 9.5 – 10.525 = –1.025 (–1.025) 2 = 1.050625 2 × 1.050625 = 2.101250
10 4 10 – 10.525 = –0.525 (–0.525) 2 = 0.275625 4 × 0.275625 = 1.1025
10.5 4 10.5 – 10.525 = –0.025 (–0.025) 2 = 0.000625 4 × 0.000625 = 0.0025
11 6 11 – 10.525 = 0.475 (0.475) 2 = 0.225625 6 × 0.225625 = 1.35375
11.5 3 11.5 – 10.525 = 0.975 (0.975) 2 = 0.950625 3 × 0.950625 = 2.851875
The total is 9.7375

The sample variance, s 2 , is equal to the sum of the last column (9.7375) divided by the total number of data values minus one (20 – 1):

s 2 = 9.7375 20 1 = 0.5125

The sample standard deviation s is equal to the square root of the sample variance:

s = 0.5125 = 0.715891 , which is rounded to two decimal places, s = 0.72.

Typically, you do the calculation for the standard deviation on your calculator or computer . The intermediate results are not rounded. This is done for accuracy.

  • For the following problems, recall that value = mean + (#ofSTDEVs)(standard deviation) . Verify the mean and standard deviation or a calculator or computer.
  • For a sample: x = x ¯ + (#ofSTDEVs)( s )
  • For a population: x = μ + (#ofSTDEVs)( σ )
  • For this example, use x = x ¯ + (#ofSTDEVs)( s ) because the data is from a sample

  1. Verify the mean and standard deviation on your calculator or computer.
  2. Find the value that is one standard deviation above the mean. Find ( x ¯ + 1s).
  3. Find the value that is two standard deviations below the mean. Find ( x ¯ – 2s).
  4. Find the values that are 1.5 standard deviations from (below and above) the mean.
    • You should get x ¯ = 10.525
    • Note this is sample data (not a population): s=0.715891
  1. ( x ¯ + 1s) = 10.53 + (1)(0.72) = 11.25
  2. ( x ¯ – 2 s ) = 10.53 – (2)(0.72) = 9.09
    • ( x ¯ – 1.5 s ) = 10.53 – (1.5)(0.72) = 9.45
    • ( x ¯ + 1.5 s ) = 10.53 + (1.5)(0.72) = 11.61

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Statistics i - math1020 - red river college - version 2015 revision a - draft 2015-10-24. OpenStax CNX. Oct 24, 2015 Download for free at http://legacy.cnx.org/content/col11891/1.8
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Statistics i - math1020 - red river college - version 2015 revision a - draft 2015-10-24' conversation and receive update notifications?

Ask