Contents

CHAPTER 11: Data Analysis and Interpretation: Part I. Describing Data, Confidence Intervals, Correlation 357sigma); the standard deviation of a sample of scores is indicated as SD whenappearing in text, but it is often symbolized as s in statistical formulas. Thevariance, a measure of dispersion that is important in the calculation of variousinferential statistics, is the square of the standard deviation, that is, s 2 .Measures of variability for the two vocabulary groups areCollegeOlder AdultRange 23–62 32–83Variance (s 2 ) 109.45 150.44Standard deviation (SD or s) 10.46 12.27Note that the stem-and-leaf display showed greater dispersion among theolder adults; with the SD we have a number to reflect that characteristic of thedistribution.Key ConceptKey ConceptStandard Error of the Mean In doing inferential statistics, we use the samplemean as a point estimate of the population mean. That is, we use a singlevalue ( X __) to estimate (infer) the population mean (). It is often helpful to beable to determine how much error there is in estimating on the basis of X __.The central limit theorem in mathematics tells us that if we draw an infinitenumber of samples of the same size and we compute X __for each of these samples,the mean of these samples means ( _ ) will be equal to the populationXmean (), and the standard deviation of the sample means will be equal to thepopulation standard deviation () divided by the square root of the samplesize (N). The standard deviation of this theoretical sampling distribution ofthe mean is called the standard error of the mean ( _ ) and is defined asX _ ____ X NTypically, we do not know the standard deviation of the population, so weestimate it using the sample standard deviation (s). Then we may obtain anestimated standard error of the mean using the formula__s X ____ s NSmall values of s __ X suggest that we have a good estimate of the population mean,and large values of s __ X suggest that we have only a rough estimate of the populationmean. The formula for the standard error of the mean indicates that ourability to estimate the population mean on the basis of a sample depends on thesize of the sample (large samples lead to better estimates) and on the variabilityin the population from which the sample was drawn, as estimated by the samplestandard deviation (the less variable the scores in a population, the better our estimateof the population mean will be). As we will show later, the standard errorof the mean plays an important role in the construction of confidence intervalsand is frequently displayed along with sample means in a figure summarizingresults of a research study.