Statistics for Dummies by Deborah Jean Rumsey

Author:Deborah Jean Rumsey [Rumsey, Deborah Jean]
Language: eng
Format: epub
Tags: Non-Fiction, Reference
ISBN: 9780764554230
Publisher: Wiley

Measuring sample variability

Sample results vary, but by how much? According to the central limit theorem (see Chapter 9), when sample sizes are large enough, the distribution of the sample proportions (or the sample averages) will follow a bell-shaped curve (or normal distribution — see Chapter 8). Some of the sample proportions (or sample averages) will overestimate the population value and some will underestimate it, but most will be close to the middle. And what's the middle? If you averaged out the results from all of the possible samples you could take, the average would be the real population proportion, in the case of categorical data, or real the population average, in the case of numerical data. Normally, you don't have the time or the money to look at all of the possible sample results and average them out, but knowing something about all of the other sample possibilities does help you to measure the amount by which you expect your one sample proportion (or average) to vary.

Standard errors are the basic building block of the margin of error. The standard error of a statistic is basically equal to the standard deviation of the data divided by the square root of n (the sample size). This reflects the fact that the sample size greatly affects how much that sample statistic is going to vary from sample to sample. (See Chapter 9 for more about standard errors.)

The number of standard errors you wish to add or subtract to get the margin of error depends on how confident you wish to be in your results (this is called your confidence level). Typically, you want to be about 95% confident, so the basic rule is to add or subtract about 2 standard errors (1.96, to be exact) to get the margin of error. This allows you to account for about 95% of all possible results that may have occurred with repeated sampling. To be 99% confident, you add and subtract about 3 standard errors (2.58, to be exact). (See Chapter 12 for more discussion on confidence levels and number of standard errors.)

TECHNICAL STUFF To be exact about the number of standard errors you wish to add or subtract in order to calculate the margin of error for any confidence level, you need to use a special bell-shaped curve called the standard normal distribution. (See Chapter 8 for details.) For any given confidence level, a corresponding value on the standard normal distribution (called a Z-value) represents the number of standard errors to add and subtract to account for that confidence level. For 95% confidence, the exact Z-value is 1.96 (which is "about" 2), and for 99% confidence, the exact Z-value is 2.58 (which is "about" 3). Some of the more commonly used confidence levels, along with their corresponding Z-values, are given in Table 10-1.

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