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GPS PreCalculus
Day 75 Notes – Normal Distributions
Normal Distribution: a probability distribution with the mean( X ) and standard devitation ( s X ) modeled by a
bell shaped curve. The term Normal Curve refers to a smooth, symmetrical bell-shaped curve.
34%
34%
13.5%
0.15%
13.5%
2.35%
X - 3s X
2.35%
X - 2s X
X - 1s X
X + 1s X
X
X + 2s X
0.15%
X + 3s X
Important info about normal distributions:
1. 50% of the data lies above the mean, 50% of the data lies below the mean
2. 68% of the data lies between -1 and +1 standard deviations from the mean
3. 95% of the data lies between -2 and +2 standard deviations from the mean
4. 99.7% of the data lies between -3 and +3 standard deviations from the mean
Standard Normal Distribution: a normal distribution with a mean = 0 and a standard deviation = 1
34%
34%
13.5%
2.35%
0.15%
Z-scores
13.5%
-3
-2
2.35%
-1
0
+1
+2
0.15%
+3
Z-score: the number z of standard deviations that a data value lies above or below the mean
z=
x- X
sX
–1
GPS PreCalculus
Day 75 Notes – Normal Distributions
Example 1: A set of data is normally distributed with a mean( X ) and standard devitation ( s X ) .
a.
P(X - s X Ј x Ј X + 2s X )
This question is asking what percentage of the data lies between -1 standard deviation and +2
standard deviations. Since it is a NORMAL DISTRIBUTION, we can use the percentages from the
normal distribution curve
So, we add up the percentages from -1 standard deviation to +2 standard deviations:
34% + 34% + 13.5% = 81.5%
Final Answer: 81.5%
b.
P( x Ј X + 2s X )
This question wants to know what percentage is less than +2 standard deviations from the mean.
Again, using our normal distribution curve, we add up all the percentages LESS THAN +2
standard deviations: 13.5% + 34% + 34% + 13.5% + 2.35% + 0.15% = 97.5%
Final Answer: 97.5%
Example 2: Math scores of an exam are normally distributed with a mean = 496 and a standard deviation = 109
a.
b.
c.
What percentage scored between 387 and 605?
What percentage scored less than 278?
Find the z-score for a test score of 630?
Solution:
First, we must draw a normal distribution and label our x-axis appropriately
34%
34%
13.5%
0.15%
169
13.5%
2.35%
2.35%
278
387
Mean-1SD:
496-109
496
Mean
605
714
0.15%
823
Mean+1SD:
496+109
Solution (a): Add up the percentages from 387-605: 34% + 34% = 68%
Solution (b): Add up the percentages less than 278: 2.35% + 0.15% = 2.5%
Solution (c): Use the formula to find a z-score: z =
x - X 630 - 496
=
= 1.2
sX
109
Assignment: Complete WS 8.2 and TURN-IN
–2