1. Normal Distribution and Z-scores
Intro to Statistics

2. What is Normal Distribution?
Modeled by a Bell-shaped curve.
Symmetric about the mean.
Mean (average) and the Median (middle #) are the same number. Or at least very close!!
Can look tall and skinny, short and wide or somewhere in the middle.

3. For a given interval on a horizontal axis, the area under the curve over that interval represents the probability that an outcome will fall within the interval. .

4. What are Standard deviations?
A measure of spread that expresses the amount of variation in the data from the mean. Symbolized by σ
Which graph has a larger spread (larger standard deviation)?
They are always
Positive!!

5. What is the Empirical Rule ?
Shows where a certain percentage of the data is found in a normal distribution.
68% of the data falls within one standard deviation of the mean.
95% of the data falls within 2 standard deviations of the mean.
99.7% (almost 100!) of the data falls within 3 standard deviations of the mean.

6. EX 1. Suppose a set of test scores for 50,000 students is normally distributed with µ = 81 and σ = Label the bell curve. Try Labeling the graph on your own!

7. What is the probability that a student scored below 76?
What is the probability that a student scored above 83.5?

8. What’s the probability that a student scored between 78.5 and 86?
What’s the probability that a student scored above 86?

9. How many students score above 86?

10. Z-scores Formula definition
𝒛= 𝒙−𝝁 𝝈 x = data point µ = mean σ= standard deviation
The measure of how many standard deviations a point is from the mean.
Positive z-score is above the mean
Negative z-score is below the mean

11. # of Std. deviations = z-score
# of Std. deviations = z-score
If a data point is 2.5σ above the mean, what is its z-score?

12. What would be an unusual z-score?
< -2
> 2
What would be an unusual z-score?

13. How do I calculate z-score (example)
Calculate the z-score for the following data points if µ = 45 ft and σ = 10 ft. X1 = 55 ft X2 = 30.ft X3 = ft.
𝒛= 𝒙−𝝁 𝝈
𝑧= 55−10 45
Z = 1
Do the next two on your own!

14. How do I find the data point if I know the z-score?
You have a Normal Distribution with mean µ = and Standard σ= Which data point has a z-score of ?
−𝟑.𝟒𝟓= 𝒙−𝟐𝟑𝟓.𝟕 𝟒𝟏.𝟓𝟖
X = 92.25
𝒛= 𝒙−𝝁 𝝈
Do the next one on your own!

15. How do we compare two sets of data?
We convert both scores to the same unit of measurement.
To do this we standardize each set by changing the mean to zero and the
Standard deviation to one.
Example: The average grade on a quiz was 41 points with a standard deviation of 3
Label both bell curves.
Normal distribution
Std. distribution
This is really just finding the z-scores!!

16. Find the z-score of each one Compare the z-scores.
Suppose your friend receives an 80% on a test in AP World History and a 90% on a test in Underwater Basket Weaving. The average score on the AP World history test was 72% with a standard deviation of The average score on the Underwater Basket Weaving test was 92% with a standard deviation of 3. Which student should be happier with their score?
Find the z-score of each one
Compare the z-scores.

17. z = 1.23 z = -.67 Example AP World x = 80 µ = 72 σ= 6.5
UBW x = 90 µ = 92 σ= 3
𝒛= 𝟖𝟎−𝟕𝟐 𝟔.𝟓
𝒛= 𝟗𝟎−𝟗𝟐 𝟑
z = 1.23
z = -.67

18. How can I find Standard deviation?
Use your calculator!!
Mean →
Standard deviation →
# of data points →

19. Scroll down to find more data!!
Median →
Remember …. Range = Max- min

20. Do the warm-up, using your calculator!
Example
Do the warm-up, using your calculator!

21. You Try!!
The Virginia Cooperative Extension reports that the mean weight of yearling Angus steers is 1150 pounds with a standard deviation of 80 pounds. 1. Label the model

22. What would be the z- score for a cow weighing 980 pounds?
What would be the z-score for a cow weighing 1340 lbs?
Which of these cows has a more surprising weight? Why?

23. How Do we find probability for an event with a non-integer z-score?
Consider the weights of 18 month old boys in the U.S. According to published growth charts, the average weight is approximately kg with standard deviation of 1.28 kg. Calculate the percentage of 18 month old boys in the U.S. who weigh between 10.5 and 14.4 kg.
Before we answer this question, get out your notesheet with the three bland bell-curves on it.

24. How to find probability using the Calculator
2ND VARS (DIST), then select #2 normalcdf Normalcdf(L,U, µ, σ) L = lower data point U = upper data point µ= mean σ = standard deviation

25. L = a and U = b L = -1E99, U = x P (a < x < b) P ( x < b)
P ( x > a)
L = a, U = 1E99 a

26. How Do we find probability for an event with a non-integer z-score?
Consider the weights of 18 month old boys in the U.S. According to published growth charts, the average weight is approximately kg with standard deviation of 1.28 kg. Calculate the percentage of 18 month old boys in the U.S. who weigh between 10.5 and 14.4 kg.

27. The useful life of a radial tire is normally distributed with a mean of 30,000 miles and a standard deviation of miles. The company makes 10,000 tires a month. What is the probability that if a radial tire is purchased at random, it will last between 20,000 and 35,000 miles?
A 94%
B 81%
C 68%
D 47%

28. The useful life of a radial tire is normally distributed with a mean of 30,000 miles and a standard deviation of 5000 miles. The company makes 10,000 tires a month. What is the probability that if a radial tire is purchased at random, it will be less than 22,000 miles?