Chapter 6:  Becoming Acquainted with Statistical Concepts

I. Three Basic Statistical Concepts

a. Describe data

b. Find the degree of relationship between two or more variables

c. Find differences between two or more groups

II. Statistics is simply an objective means of interpreting a collection of observations. 

Example:

If you were a PE teacher and you measured the Height and the standing-long-jump performance for each student in your 7^th grade class you could:

Sum all the heights and then divide by the number of students to get the average (mean) Height.

a. M = SX/N; where S means sum, X = each student’s height, and N = number of students

b. You could also test the relationships between the two variables height and standing-long-jump

c. You might hypothesize that the taller students will jump the farthest. 

d. See Figure 6.1

e. By plotting the scores of each student one can see that there tends to be a relationship and taller people tend to jump farther.

f. It is not a perfect relationship, however.

III. Pearson product moment coefficient of correlation, r

a. When two variables are unrelated their r is approximately zero. 

b. In figure 6.1 the relationship is moderate and the r = .40 to.60

c. If all the points were on a straight line the r = 1.0

IV. The t test

a. To find the differences between groups

b. Suppose you believe that weight training will improve jumping scores. 

c. You divide the class into two groups and have one group train with weights for

8 weeks to develop leg strength.

d. You want to know whether the independent variable (weight training) produces

a change in the dependent variable (standing-long-jump)

e. The t test is used to determine whether there is a significant difference between

the two groups

V. Computers used in Statistical Analysis

a. SPSS: Statistical Package for the Social Sciences

b. BIMED: Biomedical Series

c. SAS: Statistical Analysis System

d. Spreadsheet

VI. Description and Inference

a. Are not statistical techniques

b. One cannot necessarily infer a cause and effect relationship.

c. Any statistic describes a sample of participants

d. If the sample was chosen from a larger group, or population then the findings

can be inferred (generalized) to the larger group

e. The method of selecting the sample, procedures, and context is what does or

does not allow inference

V. Selecting the Sample

a. Random selection – example: 10,000 students selected at random using table

of random numbers.  Select 200 students

b. Stratified Random sampling – the population is divided (stratified) on some

characteristic before random selection – example: Freshman = 30% (60), sophomores = 30% (60), juniors = 20% (40), seniors = 20% (40)

c. Systematic Sampling – example: telephone book

d. Random Assignment – once the sample has been selected groups must be

determined by randomly assigning participants from the sample to the groups.

e. Post Hoc Justification

i. If the sample was not randomly selected from a larger population

inferences should not be made to the larger population

ii. If the sample is representative of the larger population in terms of some characteristic, i.e., average age, racial balance, socioeconomic status, etc., some justification can be made, but weakly.

VI. Difficulty of Random Sampling

a. In many studies, random sampling is not possible, or done at all.

b. Getting volunteers is the main goal.

c. Random sampling can apply to treatments

d. What we really need is a sample good enough for our purpose

e. If we do not disregard the strict restrictions of random sampling, we would

seldom generalize beyond the sample.

f. The best possible generalization statement is to say that the findings may be

“plausible” in other participants, treatments, and situations, depending on their similarity to the study characteristics.

VII. Measures of Central Tendency

a. When we have a group of scores, one number may be used to represent the group

b. Mean = SX/N ; where M is the mean, X represents the scores, N is the number of scores

c. Median = the middle score when ranked

d. Mode = the most frequent score.

e. These terms express central tendency.

VIII. Variability

a. Within the group  of scores, each individual score differs to some degree from the central tendency score.

b. Standard Deviation and Variance are measures of this degree of variability.

Standard deviation,

        ____________

s = Ö S(X-M)²/(N-1)

Standard Deviation Equation for the Calculator

                     _____________________

             s = Ö [NSX² – (SX)² / [N(N-1)]

c. Variance = s²

d. Range of scores is from the highest to the lowest, especially when median is used instead of the mean.

IX. Frequency Distribution

a. A list of all scores and the frequency of each score.

b. Grouped Frequency distribution uses intervals:

96-100

91-95

86-90

etc.                (Some data is lost)
b. Stem and Leaf Distribution

i. Similar to grouped frequency distribution but there is no loss of data.

ii. Figure 6.2

iii.Easy to rank the scores

X. Parametric and Nonparametric statistical tests

A. Parametric assumptions:

1. Population from which the sample is drawn is normally distributed on the variable of interest.

2. The samples drawn from a population have the same variances on the variable of interest

3. The observations are independent

XI.  Nonparametric is distribution free because the previous assumptions are not met.

A. If one can meet the assumptions the Parametric tests are more powerful.

B. Power means to increase the chances of rejecting a false Null Hypothesis.

C. Assumptions can be tested by using estimates of Skewness and Kurtosis.

XII. Normal Curve

a. Z-score

b. Appendix A-2

c. Assume that the assumptions are met because the assumptions are very robust to violations, meaning that the outcome of the statistical test is relatively accurate even with severe violations of the assumptions.

d. Most research in physical activity uses parametric tests.

XIII. Probability

a. What are the odds that certain things happen?

b. In a statistical test you sample from a population of participants and events. 

c. You use probability statements to describe the confidence you place in the statistical findings. 

d. Example: 

1. (p < .05) you would expect in less than 5% of the time.

2. Alpha (a)

3. The researcher establishes an acceptable level of occurrence (called alpha, a) before the study.

4. This level of chance occurrence can vary from low to high but can never be eliminated. 

5. Levels of confidence normally 0.01 or 0.05.

XIV. Type I error

a. Is to reject the null hypothesis when the null hypothesis is true. 

b. Example: a research concludes that there is a difference between two groups, but there is really not.

XV. Type II error

a. Is not to reject the null hypothesis when the null hypothesis is false.

XVI. Correcting for Type I and Type II errors

a. You can control for type I errors by setting the alpha level. 

b. The issue is, if you had to make an error, which type of error are you willing to make.

c. Example: cancer drug study; do not accept the null hypothesis if there is any chance that the drug will work (alpha = .30).

d. When experimenters set an alpha level at a specific level before the research, they will often report the probability of a chance occurrence for the specific events of the study at the level it occurred (e.g., p = 0.12)

e. It is best to report the exact level of probability associated with the statistic and then to estimate the meaningfulness or the difference or relationship.

f. The researcher should interpret the findings within the theory and hypothesis of the study.

XVII. Beta, b

a. Although the magnitude of the type I error is specified by a, you may also make a type II error, the magnitude of which is determined by Beta, b.

XVIII. Meaningfulness (Effect Size)

a. The meaningfulness of a difference between two means can be estimated in many ways,

1. Effect size is one method. 

i. ES = (M ₁– M₂ )/s

ii. This puts the difference in standard deviation units.  0.2 or less is a small ES, 0.5 is a moderate ES, and 0.8 or more is a large ES

XIX. Confidence Intervals

a. A confidence interval provides an expected upper and lower limit for a statistic at a specified probability level, usually 95% or 99%. 

b. When we choose a sample from a population we are making an estimate of the target population.

c. There will be some sampling error, which relates to how well the sample represents the target population.

XX. Standard Error

a. Represents the variability of the sampling distribution of the statistic. 

b. We could take many samples from the population and calculate the Means of each.

c. Then took the standard deviation of all the means; we would get an estimate of the standard error.

d. Or we could simply divide the sample s by the square root of the sample size.

                _____

s_m =  s /  Ö  N

e. example: If n = 30, M = 40 and s = 8;

                 ___

s_m =  8 /  Ö  30  = 1.46

f. Setting the Confidence Level . Use table A.2

g. If our sample was normally distributed we could use the value in table A.2

h. However we can’t assume that our sample is normally distributed so we must use a distribution table to account for the sample size. 

i. A.5  a t-distribution

j. example: The critical value of t for n-1 (29) df at the .05 level of significance is 2.045.

CI = sample mean ± (standard error X confidence level table value)

CI = 40 ± (1.46 X 2.045)

CI = 40 ± 2.99

k. Therefore we are 95% confident that the interval from 37.01 to 42.99 contains the population mean.

l. Confidence intervals are used in Hypothesis testing

m. If we are comparing two groups, the difference between two means would be the observed statistic. 

n. The standard error would be the standard error of the difference (which combines the standard errors of the two means). The table value would be from the t-distribution table A.5

and the df would be (n₁ –1) + (n₂ – 1) = (N-2)

XXI. Power

a. Power is the probability of rejecting the Null Hypothesis (H_o) when the H_o is false.

b.The power ranges from 0 to 1. The greater the power the more likely you are to detect a real difference.

c. The effect size curves figures 6.8 and 6.9 show sample sizes needed for certain effect sizes at .05 and .01 alpha levels.

d. The size of the sample is very influential on power; increasing with increased n.

e. Another way of looking at power is to look at the equation for the t-distribution.

Chapter 8, page 137, equation 8.3.