Measures Of Variability And Inferential Statistics
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.
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 = 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
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
Remember the following symbols:
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
Allow us to make comparisons between individuals on the same test
or different tests. Standard scores also standardize the measuring
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
How well did they perform? Plot on the normal curve.
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.
When selecting samples from the population there
is always a certain amount
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 is a direct function of the
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
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
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
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
Level of Significance
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
You test the Ho using an independent T Test
The T value of - .536 with 18 DF is not significant at the .05 level
Therefore the null Ho that there is no significant difference between private
pubic schools is accepted.
The researcher concludes that any difference is related to chance.