Variability

Measures of variability help us describe the degree of dispersion or

variability of the scores.

A= 24,24,25,25,26,26 X = 175/5 = 25

B = 16,19,22,25,28,30,35 X = 175/5 = 25

Means are the same but the scores are very different.

Range

Measure of variability

Difference between top and bottom score

Sensitive to extremes

should you count the bottom score of zero for an absent student ?

Range

Range = Hs (high score) - Ls (low score) + 1

100 - 60 = 40 + 1= 41

High score = 100 and Low score = 60

Variance and Standard Deviation

Variance and standard deviation are very strong indicators of variability.

A deviation score is represented by subtracting the individual score from the

mean

x = X mean

The Variance equals the sum of the squared deviations divided by n

To calculate the variance and standard deviation you need the following information.

Remember the following symbols:

Standard Deviation

Common measure of variability

Shows variation in scores

The smaller the deviation the closer the scores to the mean

If all scores the same standard deviation = 0

Takes into account all scores

The Standard Deviation equals the square root of the variance

Example of Standard Deviation Calculation

First take all of the scores and list them in the x column

Next calculate the mean

The second column (each score minus the mean)

The third column (each score minus the mean squared)

Total the third column

You now have all of the information needed to calculate the standard deviation.

The following is an example:

See if you can calculate the mean and standard deviation for the following

problem.

Standard Scores

Allow us to make comparisons between individuals on the same test

or different tests. Standard scores also standardize the measuring units.

x = raw score,

X = mean score,

SD = standard deviation

Calculate Z scores for Joe on a Psychology test and a Math test.

Psychology: mean = 85, SD = 10, Joe = 80

Math: mean = 70, SD = 5, Joe = 90

Psychology

Math

How well did they perform? Plot on the normal curve.

Questions

How well did Joe do compared to others on Psychology?

He scored better than what percent of the students?

For math how well did Joe do compared to the other students.

Your answer should be in a percentage.

Sampling Error

When selecting samples from the population there

is always a certain amount of error.

Sampling error is the difference between the

population parameter and the sample statistic.

e = X - µ

Sample IQ = 90 and Population IQ = 100

e = 10

The larger the sample the smaller the e

Sampling Error
Sampling error is a direct function of the Standard Deviation Sampling errors are distributed in a normal manner with an expected mean of 0 Standard Error of the Mean |

Inferential Statistics are Used to Determine Tests of Significance

In a study first make or develop a null

hypothesis (H0 )

H0 = There is no significant difference

between SAT-9 test scores of private vs.

public 9th grade students.

You randomly select 500 public and 500

private students and administer the SAT-9.

Tests of Significance

The results indicate a mean of 50 for public and a mean of 52 for private.

Is there a difference between the two groups or is any difference related to possible

error sources?

The error could be chance or sampling error.

Ho : µA = µB

Type 1 and Type II Errors

After we accept or reject the null Ho we could be correct or we could possibly make

the wrong decision.

Type I error is called an Alpha Error

Type II error is called a Beta Error

Type I unwarranted decisions

Type II Status Quo

The level of significance is the point at which the null Ho is rejected.

The value is referred to as a p value or a probability value.

P < 0.05 indicates the probability of making a type I error is less than 5 chances in 100.

In education p < 0.05 is often used.

Could be more stringent p < 0.01

Independent T -test

The null Ho states that there is no difference between public vs private school SAT

results.

You test the Ho using an independent T Test

You conclude

cf. handout

Results

The T value of - .536 with 18 DF is not significant at the .05 level of significance

Therefore the null Ho that there is no significant difference between private and

pubic schools is accepted.

The researcher concludes that any difference is related to chance.