HUNU 500

A frequency distribution is a table of rank-ordered scores that show the number of times that value occurred, or its frequency. Sometimes it is more meaningful to express frequencies as percentages of the total distribution.

A cumulative percentage is obtained by adding the percentage value for each score to all percentages that fall below that score.

A frequency distribution can be displayed as a histogram. A histogram is a bar graph composed of a series of columns, each representing one score or class interval. (p.370). It is used to show frequencies of data. Data in specified ranges are indicated by bars.

A frequency distribution can also be displayed as a cumulative frequency or cumulative distribution.

A normal distribution is one in which the scores are evenly distributed above and below the mean, and each half is a mirror image of the other.

A skewed distribution is one in which the distribution of scores is asymmetrical about the mean. The degree to which the distribution deviates from symmetry is its skewness. Distributions can be positively skewed (skewed to the right, seen when most of the scores cluster at the low end of the causing the tail of the curve to point toward the right) or negatively skewed (skewed to the left)

Normal curve refers to a bell-shaped distribution where most of the scores fall in the middle of the scale and progressively fewer fall at the extremes.

• Measures of Central Tendency
•  Mean- averaged value
•  Median- middle value
•  Mode- most common value

Percentiles describe a score’s position within a distribution by dividing the data into 100 equal portions. The location of a score on one of these portions represents its position relative to all other scores;


Quartiles divide a distribution into four equal parts, or quarters. Three quartiles exist for any data set Q1, Q2 , and Q3 corresponding to the 25th, 50th, and 75th percentile respectively. The median is the score at the 50th percentile. Distance between the first and the third quartiles is called the interquartile range.

Variance (s2) is a measure of variability around the mean. (p. 376)

Deviation score = value – mean, = X – X(mean)

• Samples with larger deviation scores will be more variable about the mean.
• The sum of the squared deviation scores is called the sum of squares (SS).
• The variance is calculated by dividing the sum of squares by n-1

Standard Deviation is the square root of the variance

Standard Error of Measurement = Standard deviation/n

Correlation- reflects the degree of association between two sets of data, or the consistency of position within the two distributions

• Correlation coefficient – is a number between -1 and +1 whose sign is the same as the slope of the line and whose magnitude is related to the degree of linear association between two variables. Correlation coefficients closer to 1 indicate a stronger association between the two variables. The important quality of correlation coefficients is not their sign, but their absolute
• value. A correlation of -0.58 is stronger than a correlation of 0.43, even though with the former, the relationship is negative.

Coefficient of determination (r2). The square of the correlation coefficient, r2, indicates the percentage of the total variance in the Y score that can be explained by the X score.

• Norm-referenced test
• A standardized testing instrument by which the test-taker's performance is interpreted in relation to the performance of a group of peers who have previously taken the same test. Although referred to as "norm-referenced tests," they are really norm-referenced interpretations of a score of a given test. Most standardized tests are norm-referenced.

• Criterion-referenced test
• Tests developed to assess performance based on an absolute criterion. Scores are derived by comparing the performance of the test-taker to a pre-specified standard or "criterion". Results are interpreted relative to a standard that represent an acceptable model or level of performance.

•  Have a very important role in describing distributions;
•  Provide criteria to use specific statistical tests i.e. parametric vs non-parametric;
•  Do not infer statistical inference but lays the foundation for the ability to perform statistics.

• They are based on
• 1) Probability (p. 387)
• o Likelihood that an event will happen.
• o Predicts what should happen not what will happen.
• 2) Sampling error: the tendency of the data from the sample to differ from the population
• o The means from a number of samples is call a sampling distribution of means.
• o Standard error of the mean is the standard deviation of the means.

•  A range of scores with specific boundaries that should contain the population mean
•  The boundaries of the CI are based on the sample mean and its standard error
•  Indicates level of confidence that mean of the population is within an interval

Null hypothesis (H0): It states that there is no difference (due to the intervention) and postulates that there will be no statistically significant difference between the groups.

• Alternative hypothesis (H1): This statement postulates that there will be a statistically significant difference between the two groups due to the intervention or the characteristics of the groups.
•  Non-directional hypothesis- does not stipulate in advance the direction of the differences between the dependable variables of the groups.
•  Directional hypothesis- makes a specific prediction about the direction and nature of the differences between the dependent variables of the groups.

• Type I error
•  reject null when it is true.
•  Eg. Conclude that treatment does make a difference when in fact it does not.

• Type II error
•  do not reject null when it is false.
•  Eg. Conclude that treatment does not make a difference when in fact it does.

•  Indicates the criterion for determining if the difference is statistically meaningful or ‘real’ rather than occurring due to chance.
•  Researchers usually select a conservative  or p-value of <0.05
•  This is often the maximal acceptable risk that the null hypothesis is not true.
•  Determines probability of making type I error i.e reject null when it is true.

•  Probability of making type II error is denoted as  or beta.
•  Power is complement of   1 – 
•  If statistical power = 80%, then probability of not committing a type II error i.e. finding a treatment effect is 80%
•  Increases when:
• o effect size increases;
• o variance of outcome decreases;
• o sample size increases;
• o or significance level is larger.

Power analysis is a procedure for estimating (a) the likelihood of committing a type II error, or (b) sample size requirements for a specific research design.

Need to perform power analysis when designing experiments to determine feasibility.

 Provides a more powerful comparison i.e. more likely to find a significant difference.

• Parametric tests can be used when the following criteria are fulfilled:
• o Variables of each group are normally distributed (the normality assumption).
• o The normal distributions have the same standard deviation (the assumption of homogeneity of variance)
• o The data is from an interval or ratio scale
• o Random samples
• o Examples of parametric tests
• - t-test
• - analysis of variance (ANOVA)
• - post-hoc tests

• Nonparametric are used in the following situations:
• o Variables are comprised of ordinal data
• o Small samples that have not satisfied normality or equality of variance.
• o Nominal or ordinal data
• o Examples of non-parametric tests
• - Mann Whitney U Test
• - Friedman two-way ANOVA by ranks

Parametric Test

• Simplest experimental design
•  2 randomly selected groups
•  1 receives control; other receives treatment

A t-test examines means and variances of each group to determine if the two groups are statistically different. Paired t-tests can be used for matched subjects or pre-post test designs

Parametric Test

•  Compares 3 or more conditions, groups, or treatments
•  Can be independent groups or repeated measures
•  Calculates an F-statistic which indicates a significant difference between at least 2 means but does not indicate which groups are different

• 1)Multiple groups design
• Tested at one time point
• Eg. How does muscle soreness after eccentric exercise differ in those who are under 20 years of age, those 21-40 years of age and those 41-60 years of age?

• 2)2-way ANOVA
• Tested at several time points
• Eg. How does muscle soreness after eccentric exercise differ in those who are under 20 years of age, those 21-40 years of age and those 41-60 years of age at different time points – baseline, 1, 2, or 3 days after the exercise stimulus?

Parametric Test

• Post-hoc Tests
•  These tests are classified as post hoc because specific comparisons are decided after performing the ANOVA
•  Some are a priori or planned comparisons before the ANOVA was performed.
• o Tukey’s honestly significant difference (very conservative. Type I error low)
• o Newman Keul’s is not as conservative
• o Bonferroni’s correction

Non-parametric Test

•  Non-parametric counterpart of two sample (unpaired) t-test
•  Used when the variables do not meet the parametric criteria
•  Does not require equal n of both groups
•  Wilcoxon’s test is used for paired data sets

Non Parametric Test

•  ANOVA by ranks
•  Also called Friedman Chi-square test
•  Nonparametric equivalent of the two-way ANOVA