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To Compute the Population Variance and Standard Deviation, find Simple Random Samples and Compare the Standard Deviations.

Example 1: The following data represent the pulse rates (beats per minute) of the nine students enrolled in a section of Sullivan’s course in introductory statistics.  Treat the nine students as a population.

(a) Compute the population variance and population standard deviation.   

 (b) Determine two simple random samples of size three, and compute the sample variance and sample standard deviation of each sample.

(c) Which samples underestimate the population standard deviation?  Which overestimate the population standard deviation?

(a.) Press STAT and select 1: Edit. Clear L1. Press STAT and highlight CALC to display the Calc Menu. Select 1:1-Var Stats and press 2^nd ENTER. The population standard deviation is, 7.67.

To calculate the population variance,, type in the value of the standard deviation at the bottom of the screen and press the key and ENTER.

The population variance is 58.83.

(b.) To generate a random sample of size 3, first number the students from 1 to 9. The TI-83 samples with replacement so you will need to generate a sample that is larger than 3. Suppose you take a sample size of 5. Press MATH and PRB. Select 5: RandInt and enter 1.9,5. Here is one possible outcome.

In this outcome we have selected 5 students: Student Numbers: 9, 8, 6, 5 and 7. Since we need only 3 students, simply select the first three: 9, 8, and 6. (Note: Random samples vary so your results will be different.)

Enter the pulse rates of the 3 Students into L2. Press STAT and highlight CALC to display the Calc Menu. Select 1: 1- Var Stats and press 2^nd ENTER. The sample standard deviation is, 8.14.

To calculate the sample variance,, type in the value of the standard deviation at the bottom of the screen and press the key and ENTER. The sample variance is 66.26.

Repeat this process to generate another random sample of size 3.

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Example 2: Use a sample of five test scores of 65, 70, 71, 75, and 95 to answer the following questions:

(a) Compute the sample standard deviation. 

(b) Suppose the professor decides to curve the exam by adding 4 points to each person’s score.  Compute the sample standard deviation of the adjusted scores.

(c) Compare the unadjusted test score’s standard deviation to the curved test score’s standard deviation.  What effect did adding 4 to each score have on the sample standard deviation? 

(d) Multiply each exam score by 2 and determine the sample standard deviation of the new test scores.

(e) What effect did multiplying the scores by 2 have on the sample standard deviation?

(a.) Press STAT and select 1: Edit. Clear L1 and enter the sample data into L1. Press STAT and highlight CALC to display the Calc Menu. Select 1:1-Var Stats and press 2^nd ENTER. The sample standard deviation,, is 11.63.

(b.) Press STAT and select 1:Edit. Clear L2. Move the cursor to the top of L2 again and press ENTER. The cursor should now be flashing on the bottom line of the screen. On this line, type in 2^nd.

Press ENTER. Each value in L2 should be 4 points higher than the corresponding value in L1. Press STAT and highlight CALC to display the Calc Menu. Select 1:1-Var Stats and press 2^nd ENTER. The sample standard deviation,, is 11.63, the same as the standard deviation for the original data.

(d.) Press STAT and select 1:Edit. Clear L3. Move the cursor to the top of L3 again and press ENTER. The cursor should now be flashing on the bottom line of the screen. On this line, type in 2^nd Press STAT and highlight CALC to display the Calc Menu. Select 1:1-Var Stats and press 2^nd ENTER. The sample standard deviation,, is 23.26, which is twice the value of the standard deviation for the original data.