Data Matters with SPSS®

Activity 7.2

If you had only the sample means available, could you figure out the standard deviation of the population they came from?

To get you started, here is the relationship between the standard deviation of sample means (the standard error) and the standard deviation of the population from which the observations were drawn.

It would seem that you could multiply both sides by the square root of the sample size and get

To put it the other way,

Does that work? If you have only the sample size and some sample means, can you accurately estimate the population standard deviation?

To find out, you will use the same approach you used in Section 7.1: You will use a Syntax program to create samples, then you’ll aggregate to get the sample means.

To create 1,000 samples with nine observations each, set the number of samples in LOOP #Sample = 1 TO 1000. and set the sample size in LOOP #Case = 1 to 9 .

The population that the observations are drawn from has a standard deviation of 7. The population standard deviation is set in COMPUTE z = RV.NORMAL(0,7). . Set the number of samples, the sample size, and the standard deviation where you like.

INPUT PROGRAM.
  LOOP #Sample = 1 TO 1000.
    LOOP #Case = 1 to 9.
      COMPUTE sample = #Sample.
      COMPUTE measure = RV.NORMAL(0,7).
      END CASE.
    END LOOP.
  END LOOP.
  END FILE.
END INPUT PROGRAM.
EXECUTE.

Use Aggregate to get the sample means: In the data editor, click on Data, Aggregate. Sample is the break variable. Put measure into the Aggregate Variable(s) box. Click on Replace working data, OK.

Use Explore to get the standard deviation of the sample means.

The standard deviation of the sample means is an estimate of the standard error. Multiply that by the square root of the sample size to estimate the population standard deviation.

How did you do? Was your estimate a good estimate of the standard deviation of the population that the data came from?

Does it matter what the sample size is? Does it matter how many samples you work with?

What if different samples had different sizes? Could you use this equation if different samples had different sizes?