Name: __________________
Normal Distribution IPad Notes
Mean and Standard Deviation
Background: Go to the Normal Distribution page on the Sketchpad program. The graph is a picture of a
normal curve. This represents a set of data with a Mean value of 0 and a Standard Deviation value of 1. The
graph can be changed with the mean (  ) and standard deviation (  ) sliders.
1) What happens to the graph when the mean is changed to 3?
2) What happens to the graph when the mean is changed to -1?
3) Look at the equation for the normal curve.
y
e
 ( X   )2
2 2
 2
. Explain why changing  results in your
described transformation.
4) Move  back to 0. What happens to the graph when the standard deviation is changed to 2?
5) What happens to the graph when the standard deviation is changed to .5?
6) Change the mean of the graph to -3 and the standard deviation to .75. Graph the curve of the data on the
same line as the normal curve.
5
4
3
2
1
0
1
2
3
4
5
Using the above investigations, write the definitions of mean and standard deviation in your own words.
Mean (  ) Standard deviation (  ) -
Consider the following sets of data:
Period 1 Test Scores: 76, 78, 81, 81, 87, 89, 92
Period 2 Test Scores: 65, 70, 71, 85, 96, 98, 100
7. Which class had the higher mean?
8. Which class had the higher standard deviation? Explain your answer.
The Empirical Rule
Background: Standard deviation can be used to find percentages of data that is Normally Distributed. For
example, the average height of adult men is 70 inches with a standard deviation of 3 inches. The graph below
shows the normal curve representing the situation.
Adult Male Height (in inches)
  3
61"
  2
64"
  1
67"

70"
  1
73"
  2
76"
  3
79"
Use the Empirical Rule tab of the Setchpad program. Shade in the percent regions of the adult male population
on the above graph.
1) If 1000 men have their heights measured, how many men would be between 1 standard deviation below and
1 standard deviation above the average?
2) If 1000 men have their heights measured, how many men would be between 2 standard deviations above and
3 standard deviations above the average?
3) If 1000 men have their heights measured, how many men would be between 3 standard deviations below and
1 standard deviation above the average?
Z-Scores
Background: Go to the Z-Scores tab of the Sketchpad program. The z value is actually the number of standard
deviations that a particular X value is away from the mean.
1) Slide the Z score at the bottom of the graph to show a z-score of 0. What percentage is shaded to the left of
the z-score?
2) If the z-score is negative, what can you say about the percentage of data to the left of the z-score?
3) If the z-score is positive, what can you say about the percentage of data to the left of the z-score?
4) Where on the graph is a z-score of 1.11? What percentage is shaded to the left of the z-score?
5) Where on the graph is a z-score of -.65? What percentage is shaded to the right of the z-score?
Background: Go to the Z-Score Intervals tab of the Sketchpad program.
6) Find the percentage under the curve between z = 0 and z = 1. Does it match the percentage from the
empirical rule?
7) Find the area under the curve between z = 0 and z = -1.75.
8) Find the area under the curve between z = 2.00 and z = 2.47.
9) Find the percentage to the right of z = 2.43 and to the left of z = -3.01.
Probabilities
Background: The area under the curve corresponds to the probability. Problems of this type are solved in the
same manner as the previous examples. For probabilities a special notation is used. For example to find the
probability of any z value between 0 and 2.32 we would write P( 0 < z < 2.32 ).
Use either of the z-score tabs to solve the following probability problems.
1) Find: P ( 0 < z < 2.32 )
2) Find: P ( z < 1.65 )
3) Find: P ( z > 1.91 )
4) Find the z value such that the area under the normal curve between 0 and the z value is 0.2125.
P(0  z  t )  .2125
5) Find the z value to the right of the mean so that 53.98% of the area under the distribution curve lies to the left
of it. P ( Z < t ) = .5398
6) Find the z value to the left of the mean so that 98.87 % of the area under the distribution curve lies to the
right of it. P ( Z > t ) = .9887
7) Find two pairs of z values so that 40% of the middle area is bounded by them.
Calculating the Z-Scores
Background: To solve problems by using the standard normal distribution, transform the original variable to a
standard normal distribution variable by using the formula,
z
value - mean
standard deviation
or
z
X 

For example, suppose the scores for a standardized IQ test are normally distributed. The mean of the IQ test
scores is 100 with a standard deviation of 15. N (  , )  N(100, 15).
1) What is the corresponding z score for and individual that scored 120?
2) If you scored 120, you would have scored better than what percent of the individuals who took the test?
3) What is the corresponding z score for an individual that scored 75?
4) If you scored 75, what percent of the individuals who took the test scored better than you?
Battery lifetime is normally distributed for large samples. The mean lifetime is 500 days and the standard
deviation is 61 days. N (  , )  N(500, 61).
5) What percent of batteries have lifetimes less than 500 days?
6) What percent of batteries have lifetimes shorter than 550 days?
7) What is the probability of getting a battery that has a lifetime longer than 400 days?
8) What is the probability of getting a battery with a lifetime between 475 and 525 day?
Name:________________________
Normal Distribution Homework
1. Consider the sets of data of class grades for first quarter.
2nd Period: 85%, 92%, 88%, 81%, 79%, 95%
5th Period: 72%, 97%, 84%, 96%, 67%, 81%
a) Which class had a higher class grade average?
b) Which class had a higher standard deviation? Explain your reasoning.
2) Graph each of the z-scores on the normal distribution graph and shade the area indicated in the
probability. Find the probability using a graphing calculator.
a) P ( 0 < z < 1.51 )
Calculating probabilities between z-scores:
P = _________________
b) P ( -2 < z < .34 )
P = _________________
1) Go to Distribution (2nd Vars)
2) Choose #2 NormalCDF
3) Type in the lower z-score
4) Type in a comma (next to the parentheses)
5) Type in the upper z-score
6) Press enter
c) P ( z < -.78 )
Calculating probabilities less than a z-score:
1) Go to Distribution (2nd Vars)
2) Choose #2 NormalCDF
3) Type in -10
4) Type in a comma (next to the parentheses)
5) Type in the z-score
6) Press enter
P = _________________
d) P ( z < 2.14 )
P = _________________
e) P ( z > -1.67)
Calculating probabilities less than a z-score:
1) Go to Distribution (2nd Vars)
2) Choose #2 NormalCDF
3) Type in the z-score
4) Type in a comma (next to the parentheses)
5) Type in 10
6) Press enter
P = _________________
f) P ( z > 1.59)
P = _________________
3) The average admission charge for a movie is $5.39. If the std dev is $0.79, what is the probability that a
randomly selected admission charge is less than $3.00?
a) First find the z-score for $3.00. z 
X 

b) Find the probability that the admission charge is less than you z-score.
4) The average salary for a first-year teacher is $27,989 with a standard deviation of $3250. What is the
probability that a first year teacher makes between $20,000 and $30,000 a year?
a) First find the z-score for $20,000. z 
b) First find the z-score for $30,000. z 
X 

X 

c) Find the probability that the admission charge is between your z-scores.
5) A brisk walk at 4 mph burns an average of 300 calories per hour. If the standard deviation is 8 calories, find
the probability that a person who walks 1 hour at the rate of 4 mph will burn these calories.
a) More than 280 calories
b) Less than 293 calories
c) Between 285 and 320 calories
Find the z value to the right of the mean so that 53.98% of the area under the distribution curve lies to the left of
it. P ( Z < t ) = .5398
6) Find each z-score using a graphing calculator. Graph the shaded region on the normal curve.
a) Find the z value to the right of the mean so that 71.84% of the area under the distribution curve lies to the
left of it. P ( Z < t ) = .7184
Calculating z-scores from percentages:
z = __________
1) Go to Distribution (2nd Vars)
2) Choose #3 InvNorm
3) Type in the percent as a decimal
4) Press enter
**Note: This works for percents to the left of
the z-score.
b) Find the z value to the right of the mean such that 14.56% the area under the normal curve is between 0 and
the z value. P ( 0 < Z < t ) = .1456
z = __________
c) Find the z value to the left of the mean so that 62.58 % of the area under the distribution curve lies to the
right of it. P ( Z > t ) = .6258
z = __________