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Unit51.pdf
Unit 5: The Normal Curve
Learning Objectives
Define and explain the concept of the normal curve.
Convert empirical scores to Z scores and use Z scores and the normal curve table
(Appendix A) to find areas above, below, and between points on the curve.
Express areas under the curve in terms of probabilities.
Unit Outline
Using Statistics
Properties of the Normal Curve
Using the Normal Curve
Using the Normal Curve to Estimate Probabilities
Using Statistics
Using Statistics
Properties of the Normal Curve
Using the Normal Curve
Using the Normal Curve to Estimate Probabilities
Properties of the normal Curve
The normal curve is a theoretical model.
The normal curve is unimodal, perfectly smooth, and symmetrical; the mode, median,
and mean are the exact same value.
The normal curve is bell-shaped and has “tails” that extend infinitely in both directions.
Distances along the horizontal axis are measured in standard deviation units.
When measured in standard deviations from the mean, distances along the horizontal axis
will always encompass the same proportion of the total area under the curve.
The proportion of cases falling between the mean and one standard deviation
above the mean on one variable will be the exact same as the proportion of cases
falling between the mean and one standard deviation above the mean on a
different variable.
68.26% of the area under the normal curve is found between 1 standard deviation below
the mean, and 1 standard deviation above the mean.
95.44% of the area under the normal curve is found between 2 standard deviations below
the mean, and 2 standard deviation above the mean.
99.72% of the area under the normal curve is found between 3 standard deviations below
the mean, and 3 standard deviation above the mean.
Areas under the normal curve can be expressed as the proportion of area, the
number of cases, or the probability of a case, falling above some point, below some
point, or between two points.
Using the Normal Curve
To use the normal curve, we calculate Z scores.
Z scores are “raw” values that have been “standardized” (or converted) into
standard deviation units.
A distribution of Z scores will have a mean of zero (0) and a standard deviation of
one (1).
Computing Z- Scores
Z scores are measured in units of standard deviation.
A score that falls one standard deviation above the mean will have a Z
score of +1.
A score that falls two standard deviations below the mean will have a
Z score of -2.
Z scores can take on fractional values; for example, a score can fall
2.75 standard deviations above the mean.
s
xx Z i
The normal curve table presents all possible areas under the normal curve for each
Z score.
For each Z score, the table provides the proportion of the area:
Between the mean and that particular Z score (the B Column)
From that Z score and away from the mean (the C Column)
Because the normal curve is symmetrical, the normal curve table only includes
positive Z scores. The areas for negative Z scores will be identical.
The total area above a mean is 0.50 (50%); the total area below a mean is 0.50
(50%). The total area under the entire curve is 1.00 (100%).
The normal curve table presents all possible areas under the normal curve for each Z
score.
For each Z score, the table provides the proportion of the area:
Between the mean and that particular Z score (the B Column)
From that Z score and away from the mean (the C Column)
Because the normal curve is symmetrical, the normal curve table only includes positive Z
scores. The areas for negative Z scores will be identical.
The total area above a mean is 0.50 (50%); the total area below a mean is 0.50 (50%).
The total area under the entire curve is 1.00 (100%).
Finding the Total Area Above or Below a Score
a. Calculate the Z score.
b. Find the Z score in the normal curve table.
c. Use the B and C columns of the table to determine the area.
Finding the Area Below a Positive Z Score*
In a distribution with a mean of 100 and a standard deviation of 20,
find the area below 108 by:
a) calculating the Z score (.40) (see below);
b) finding the Z score in the normal curve table; &
c) reporting the area in the B Column + .50
(.1554+.50=.6554).
Multiply by 100 to convert your answer to a
percentage (65.54%)
*scores above the mean have a Positive Z Score
Finding the Area Below a Negative Z Score*
In a distribution with a mean of 100 and a standard deviation of 20,
find the area below 80 by:
a) calculating the Z score (-1.00) (see below);
b) finding the Z score in the normal curve table; &
c) reporting the area in the C Column (.1587)
Multiply by 100 to convert your answer to a
percentage (15.87%)
*scores below the mean have a Negative Z Score
Finding Areas Between Two Scores
a. Calculate the Z score for both scores.
b. Find the Z scores in the normal curve table.
Use the B and C columns of the table to determine the area.
Finding the area between a Negative Z Score and a Positive Z Score.
In a distribution with a mean of 100 and a standard deviation of 20,
find the area between 93 and 112 by:
a) calculating the Z scores (-.35 and .60) (see below);
b) finding the Z scores in the normal curve table; &
c) reporting the sum of the areas in the B Columns
(.1368+ .2257 = .3625)
Multiply by 100 to convert your answer to a percentage (36.25
40. 20
8
20
100108
s
XX Z t
00.1 20
20
20
10080
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XX Z t
Finding the area between two Negative Z Scores or two Positive Z Scores.
In a distribution with a mean of 100 and a standard deviation of 20,
find the area between 113 and 121 by:
a) calculating the Z scores (+.65 and +1.05) (see below);
b) finding the Z scores in the normal curve table; &
c) reporting the difference of the areas in the B Columns
(.3531 - .2422 = .1109) or the C Columns (.2578 - .1469 = .1109)
Multiply by 100 to convert your answer to a percentage (11.09%)
Using the Normal Curve to Estimate Probabilities
The normal curve can also describe the probability (or likelihood) that a score will
fall above, below, or between scores in a distribution.
The same steps used in the previous four slides can be used to calculate
probabilities that a value will fall above a score, below a score, or between two
scores.
Note that the probability of randomly selecting a value close to the mean (+/- 1
standard deviation) is greater than that of selecting a value further away from the
mean.
Summary
The normal curve can also describe the probability (or likelihood) that a score will
fall above, below, or between scores in a distribution.
The same steps used in the previous four slides can be used to calculate
probabilities that a value will fall above a score, below a score, or between two
scores.
Note that the probability of randomly selecting a value close to the mean (+/- 1
standard deviation) is greater than that of selecting a value further away from the
mean.
