COURSE: JUST 3900INTRODUCTORY STATISTICS

FOR CRIMINAL JUSTICE

Instructor:Dr. John J. Kerbs, Associate Professor

Joint Ph.D. in Social Work and Sociology

Chapter 7: Chapter 7: The Distribution of Sample MeansThe Distribution of Sample Means

Samples and PopulationsSamples and Populations Samples provide an incomplete picture of the Samples provide an incomplete picture of the

population.population. There are aspects of the population that may not be There are aspects of the population that may not be

included within a sample.included within a sample. The The sampling error sampling error is the natural discrepancy (i.e., is the natural discrepancy (i.e.,

the the differencedifference), or amount of error, between a sample ), or amount of error, between a sample statistic and its corresponding population parameter. statistic and its corresponding population parameter. The sampling error is the measure of the The sampling error is the measure of the

discrepancy (i.e., discrepancy (i.e., differencedifference) between the sample ) between the sample and the population.and the population.

A Sampling DistributionA Sampling Distribution

A A Sampling Distribution Sampling Distribution is a distribution of is a distribution of statistics obtained by selecting all of the statistics obtained by selecting all of the possible samples of a specific size (n) from possible samples of a specific size (n) from a population.a population.

The Distribution of Sample MeansThe Distribution of Sample Means

The The Distribution of SampleDistribution of Sample MeansMeans is defined is defined as the set of sample means for all of the as the set of sample means for all of the possible random samples of a particular size possible random samples of a particular size ((nn) that can be selected from a specific ) that can be selected from a specific population. population. Often called the Often called the Sampling Distribution of Sampling Distribution of MMThis distribution has well-defined (and predictable) This distribution has well-defined (and predictable)

characteristics that are specified in the Central characteristics that are specified in the Central Limit Theorem Limit Theorem

The Distribution of Sample MeansThe Distribution of Sample Means The three characteristics of the The three characteristics of the Distribution of SampleDistribution of Sample

MeansMeans 1. Sample means should pile up around the population 1. Sample means should pile up around the population

meanmean 2. The pile of sample means should tend to form a normal-2. The pile of sample means should tend to form a normal-

shaped distribution. They should pile up in the center of shaped distribution. They should pile up in the center of the distribution (around the distribution (around μμ) and the frequencies should ) and the frequencies should taper off as the distance between taper off as the distance between MM and and μ μ increases. increases.

3. In general, the larger the sample, the closer the sample 3. In general, the larger the sample, the closer the sample means should be to the population mean (means should be to the population mean (μμ).).

Larger samples are more representative of the population than Larger samples are more representative of the population than smaller samplessmaller samples

Sample means obtained with large samples (i.e., a large Sample means obtained with large samples (i.e., a large nn) should ) should cluster relatively close to the population parametercluster relatively close to the population parameter

Means obtained by small samples should be more widely scatteredMeans obtained by small samples should be more widely scattered

The Central Limit TheoremThe Central Limit TheoremThe Central Limit Theorem is defined as The Central Limit Theorem is defined as

follows:follows:For any population with a mean For any population with a mean ((μμ)) and and

standard deviation standard deviation ((σσ)), the distribution of , the distribution of sample means for sample size sample means for sample size nn will have a will have a mean of mean of μμ and a standard deviation of and a standard deviation of σσ and will approach a normal distribution as and will approach a normal distribution as nn approaches infinity approaches infinity (∞)(∞)..

The Central Limit TheoremThe Central Limit Theorem

1. The 1. The Expected Value of Expected Value of MM is the mean of the distribution is the mean of the distribution of sample means and the Expected Value of of sample means and the Expected Value of MM is is always equal to the mean of the population of scores always equal to the mean of the population of scores ((μ)μ)..

2. The shape of the distribution of sample means tends to 2. The shape of the distribution of sample means tends to be normal. It is guaranteed to be normal if either be normal. It is guaranteed to be normal if either a) the a) the population from which the samples are obtained is population from which the samples are obtained is normalnormal, or , or b) the sample size b) the sample size nn ≥ 30 ≥ 30..

3. The standard deviation of the distribution of sample 3. The standard deviation of the distribution of sample means is called the means is called the Standard Error of Standard Error of MM ( (σσMM) ) and is and is computed by the following: computed by the following:

The Expected Value of MThe Expected Value of M

If two (or more) samples are selected from the If two (or more) samples are selected from the same population, the two samples probably same population, the two samples probably will have different means. will have different means.

Although the samples will have different Although the samples will have different means, you should expect the sample mans to means, you should expect the sample mans to be close to the population mean.be close to the population mean.

The mean of the distribution of the sample of The mean of the distribution of the sample of means is means is equalequal to the mean of the population to the mean of the population of scores (μ): that is the of scores (μ): that is the expected value of expected value of MM..

Standard Error of MStandard Error of M The The standard error standard error (also known as the standard deviation of (also known as the standard deviation of

the distribution of sample means, the distribution of sample means, σσMM) provides a measure of the ) provides a measure of the

average distance between average distance between MM (sample mean) and μ (population (sample mean) and μ (population mean).mean).

σσM M describes the distribution of sample means (variability)describes the distribution of sample means (variability)

σσM M shows how much distance is expected between shows how much distance is expected between MM and μ and μ Law of large numbersLaw of large numbers: The larger the sample size (n), the : The larger the sample size (n), the

more probable or likely it is that more probable or likely it is that MM is close to μ. is close to μ. Inverse relationship: the larger the sample size, the smaller Inverse relationship: the larger the sample size, the smaller

the stander error.the stander error. Small standard errors indicate that sample means are close Small standard errors indicate that sample means are close

together (large standard errors indicate that means are together (large standard errors indicate that means are scattered over a large range with larger difference from one scattered over a large range with larger difference from one sample to another)sample to another)

The Standard Error of MThe Standard Error of M

The standard error of The standard error of MM is defined as the is defined as the standard deviation of the distribution of standard deviation of the distribution of sample means and measures the standard sample means and measures the standard distance between a sample mean and the distance between a sample mean and the population mean. population mean.

Thus, the Standard Error of Thus, the Standard Error of M M provides a provides a measure of how accurately, on average, a measure of how accurately, on average, a sample mean represents its corresponding sample mean represents its corresponding population mean.population mean.

The Standard Error of MThe Standard Error of MConsider the changes in Standard

Error of M as n increases from 1 to 4 and then to 100 for a normal

population with a mean of 80 (μ=80) and a standard deviation of 20 (σ=20)

Do NOT confuse “standard deviations”

with“standard errors”

Difference Between Standard Deviations Difference Between Standard Deviations and Standard Errorsand Standard Errors

Standard Deviation measures the distance between Standard Deviation measures the distance between a scorea score and and the population meanthe population mean X - X - μμ

The Standard Error measures the distance between The Standard Error measures the distance between a sample a sample mean mean and and the population meanthe population mean M – M – μμ

The Standard Error (The Standard Error (σσMM) is the same as the Standard ) is the same as the Standard

Deviation for Deviation for nn = 1 = 1 Note: there is only one population meanNote: there is only one population mean

Probability and Sample MeansProbability and Sample Means

Because the distribution of sample means Because the distribution of sample means tends to be normal, the tends to be normal, the zz-score value obtained -score value obtained for a sample mean can be used with the unit for a sample mean can be used with the unit normal table to obtain probabilities. normal table to obtain probabilities.

The procedures for computing The procedures for computing zz-scores and -scores and finding probabilities for sample means are finding probabilities for sample means are essentially the same as we used for individual essentially the same as we used for individual scoresscores

Probability and Sample Means Probability and Sample Means (cont'd.)(cont'd.)

However, when you are using sample means, However, when you are using sample means, you must remember to consider the sample you must remember to consider the sample size (n) and compute the standard error (size (n) and compute the standard error (σσMM) ) before you start any other computations. before you start any other computations.

Also, you must be sure that the distribution of Also, you must be sure that the distribution of sample means satisfies at least one of the sample means satisfies at least one of the criteria for normal shape before you can use criteria for normal shape before you can use the unit normal table:the unit normal table:1. the population from which the samples are 1. the population from which the samples are

obtained is normal, or obtained is normal, or 2. the sample size (2. the sample size (nn) is 30 or more.) is 30 or more.

zz-Scores and Location within the -Scores and Location within the Distribution of Sample Means Distribution of Sample Means

Within the distribution of sample means, the Within the distribution of sample means, the location of each sample mean can be specified location of each sample mean can be specified by a by a zz-score:-score:

((MM – – μ)μ)zz = = ────────── or or

σσMM

((MM – – μ)μ)zz = = ──────────

((σ/σ/√√n)n)

zz-Scores and Location within the -Scores and Location within the Distribution of Sample Means Distribution of Sample Means

(Continued)(Continued)As always, a As always, a positivepositive zz-score-score indicates a indicates a

sample mean that is sample mean that is greater than greater than μμ and a and a negativenegative zz-score-score corresponds to a sample corresponds to a sample mean that is mean that is smaller than smaller than μμ. .

The numerical value of the The numerical value of the zz-score indicates -score indicates the distance between the distance between M M and and μμ measured measured in terms of the standard errorin terms of the standard error..

Distribution for Sample Means Distribution for Sample Means (n(n = 25, = 25, μμ = 500, = 500, σσ = 100 = 100))

A score of 540 is two standard errors above

the mean (z=+2.00), which is very unlikely (see Unit Normal Table

for z = +2.00, p = 0.0228)

2.28%

More Thoughts on More Thoughts on Standard ErrorStandard Error

Standard errors are nothing more than measures of reliability.Standard errors are nothing more than measures of reliability. Vogt (2005, p. 274) definesVogt (2005, p. 274) defines reliabilityreliability as follows: as follows:

Freedom from measurement (random) error. In practice, Freedom from measurement (random) error. In practice, this boils down to consistency or stability of a measure or this boils down to consistency or stability of a measure or test or observation from one use to the next. When test or observation from one use to the next. When repeated measures of the same thing give highly similar repeated measures of the same thing give highly similar results, the measurement instrument is said to be reliable.results, the measurement instrument is said to be reliable. Small standard errors indicate that sample means are

close together and so researchers can be fairly confident that an individual sample mean can act as a reliable measure of the population mean

Large standard errors indicate problems with reliability