The Normal Distribution: A Statistical Concept Explained, Exams of Statistics

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The Normal Distribution

Course: Statistics 1

Lecturer: Dr. Courtney Pindling

The Normal Distribution

Family of Normal

Distributions

– Mean and Standard Deviations

Symmetrical , bell-

shaped , and unimodal

distribution

Represents distributions of

continuous variables

An assumption of many

inferential statistical

methods

The Standard Normal Distribution

The Standard Normal

Distribution is the

distribution of z-scores

Area under curve is

1 or 100%

Cumulative percent z

– % area from left to z-value

Area between z-scores:

– 68% between z = -1 and z = +

– 95% between z = -2 and z = +

– 99% between z = -3 and z = +

Normal Curve and z-score

Proportion (Percentage) Under Curve

• Between pairs of z-scores
• Less than specified z-score
• Greater than specified z-score

Percentile, Ppercent value

• The raw score that a given % of distribution is less than or equal to

Percentile Rank, PRX

• The percentile of a given raw score

Normal Curve Equivalent (NCE)

– A normalized standard score; NCE = 21Z + 50

Percent when z = 1

From cumulative z

table

Z= 1 yields 0.

Convert to Percent by

multiplying by 100

84.13% when z = 1

So 84.13% of scores

are =< z = 1

Area Under Curve

Cumulative z-score Table

Proportion or Percentage of z-score

– Example 1: Percent < z = 0

Table z = 0 is 0.5, so 50%

– Example 2: Percent < z = 1

Table z = 1 is 0.8413, so 84.13%

– Example 3: Percent < z = 1.

Table z = 1.65 is 0.9505, so 95.05%

– Example 4: Percent > z = 1

Since 84.13% < z = 1, so 15.87% ( 100 – 84.13)

– Example 5: Percent between z = 1 and z = 1.

Percent Rank, PR

X

Cumulative z-score Table

– The percentile of a given raw score

– Example 6: Percent Rank of X = 82

Calculated z-score is -1.

Table z = -1.48 is 0.0694, so % < z = -1.48 = 6.94%

PR 82 is 6.94%

– Example 7: Percent Rank of X = 89

Calculated z-score is +1.

Table z = 1.32 is 0.9066, so % < z = 1.32 = 90.66%

PR 89 is 90.66%

Z-score of a Percentage

What value of z corresponds to (< %):

– 45%? Find 0.45 proportion in cumulative table

and locate its z-score

– Its z-score is between z = -0.12 and -0.

– So the 45% point represents a z score of about

z 0 0.01 0.02 0.03 0.

-0.1000 0.4602 0.4562 0.4522 0.4483^ 0.

-0.2000 0.4207 0.4168 0.4129 0.4090^ 0.

Normal Curve Equivalent - NCE

A way of measuring where a student score

falls along the normal curve

NCE scores run from 1 to 99

Standard Score: M = 50, SD = 21.

Similar to Percent Rank (1 to 100)

NCE can be averaged

Good for measuring school-wide gains and

losses in student achievement

NCE = 21Z + 50

NCE Comparisons