achievement tests reference

8/20/2019 Achievement Tests Reference

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Normal distributions (bell shaped) are a

family of distributions that have the

same general shape. They are

symmetric (the left side is an exactmirror of the right side) with scores

more concentrated in the middle than

in the tails. Examples of normaldistributions are shown to the right.

Notice that they differ in how spreadout they are. The area under each

curve is the same.

If your data fits a normal distribution, approximately 68% of your subjects will fall

within one standard deviation of the mean.

Approximately 95% of your subjects will fall within two standard deviations of

the mean.

Over 99% of your subjects will fall within three standard deviations of the

mean.


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The mean and standard deviation are useful ways to describe a set of

scores. If the scores are grouped closely together, they will have a smaller

standard deviation than if they are spread farther apart.

Small Standard Deviation Large Standard Deviation

Different Means

Different Standard Deviations

Different Means

Same Standard Deviations

Same Means

Different Standard Deviations

When you have a subject’s raw score, you can use the mean

and standard deviation to calculate his or her standardizedscore if the distribution of scores is normal. Standardized

scores are useful when comparing a student’s performance

across different tests, or when comparing students with each

other.

z-score -3 -2 -1 0 1 2 3

T-score 20 30 40 50 60 70 80

IQ-score 65 70 85 100 115 130 145

SAT-score 200 300 400 500 600 700 800


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The number of points that one standard deviation equals varies from distribution to

distribution. On one math test, a standard deviation may be 7 points. If the meanwere 45, then we would know that 68% of the students scored from 38 to 52.

24 31 38 45 52 59 63

Points on Math Test

30 35 40 45 50 55 60

Points on a Different Test

On another test, a standard

deviation may equal 5 points. If the

mean were 45, then 68% of thestudents would score from 40 to 50points.

Length of Right Foot

8

76

54

3

21

4 5 6 7 8 9 10 11 12 13 14

Data do not always form a normal distribution. When most of the scores are high,

the distributions is not normal, but negatively (left) skewed.

N u m b e r o

f P e o p l e w i t h

t h a t S h o e S i z e

Skew refers to the tail of the distribution.

Because the tail is on the negative (left) side of the graph, the distribution has a

negative (left) skew.


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N u m b e

r o f P e o p l e w i t h

t h a t S h

o e S i z e

When most of the scores are low, the distributions is not normal, but positively

(right) skewed.

Because the tail is on the positive (right) side of the graph, the distribution has a

positive (right) skew.

When data are skewed, they do not possess the characteristics of the

normal curve (distribution). For example, 68% of the subjects do not fallwithin one standard deviation above or below the mean. The mean, mode,

and median do not fall on the same score. The mode will still berepresented by the highest point of the distribution, but the mean will be

toward the side with the tail and the median will fall between the mode and

mean.

Negative or Left Skew Distribution Positive or Right Skew Distribution


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