Statistics:

Descriptive

Individuals

Relationships

Test

Items

Descriptive Statistics

Maximum

The biggest number in a set of scores.

Minimum

The smallest number in a set of scores.

Range

Range = Maximum - Minimum

Mean

The arithmetic average of a group of scores.

 Mean =  where,
 

x	=	sum of all scores, and
n	=	the number of scores.

Median

The midpoint of a range of scores.

 Median =  where,
 

l	=	exact lower limit of the interval containing the median,
	=	number of people below l
	=	number of people in the interval containing the median, and
i	=	size of the interval.

Standard Deviation

Provides a reference of a group of scores to the normal curve or, it describes the variability in a group of scores.

SD =  where,
 

x	=	a score,
	=	mean,
n	=	number of scores, and
	=	sum of.

Test Statistics for Individuals

z-score

Indicates where a person places on the normal curve.

z =  where,
 

x	=	individuals score,
	=	mean of all scores, and
SD	=	standard deviation

T score

Indicates a persons score such that an average score is 50 and for every 10 points above or below 50 indicates 1 standard deviation above or below the average of the group.

 T = 10z + 50 where,
 

z

=

individuals z-score

Stanine

The stanine, or standard nine, indicates a persons score such than an average score is 5 and for every 2 points above or below 5 indicates 1 standard deviation above or below the average of the group.

 

Stanine = 2z + 5 where,
 

z

=

individuals z-score.

Statistic of Relationship

Pearson Product Moment Correlation

A number ranging from -1.00 to +1.00 that indicates how well two groups of numbers relate to each other. +1.00 equals a perfect positive relationship, -1.00 equals a perfect negative relationship, and 0.00 indicates no relationship.

r =  where,
 

	=	mean of the group of x scores,
	=	mean of the group of y scores,
n	=	number of xy pairs,
	=	sum of all pairs of x score multiplied by y score,
	=	standard deviation of x scores,
	=	standard deviation of y scores.

Statistics About a Test

Reliability

Indicates the consistency with which a test measures. That is, it indicates how consistent the scores are in two different administrations of the same test.

Split-Half Reliability

1. Compute the Pearson Product Momment Correlation between the scores obtained for each person on the odd items and the scores obtained on the even items.
2. Correct the scores using the Spearman-Brown Prophecy Formula.

Spearman-Brown Prophecy Formula

where,

 

K	=	ratio to increase to desired length of a test (usually 2), and
r	=	odd even correlation of scores.

Cronbach's Coefficient alpha Reliability

where,

 

k	=	number of items in test,
	=	standard deviation of item, and
	=	standard deviation of test.

Statistics About Test Items

Item Difficulty

Indicates how difficult an item is. Ranges between zero, no one got the item correct, to one, everyone got the item correct.

where,
 

n	=	number of people who took the test, and
	=	number of people who got the item correct.

Item Discrimination

Indicates how well the item distinguished between those who did well on the test and those who did not.

1. Correlate the peoples' responses on the item and the peoples' score on the test.