1. Normal Distribution and Z-scores
Intro to Statistics

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

3. 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)?

4. 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.

5. If µ = 70 and σ = 3, What percent of the class scored between 67 and 73?

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

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

8. Do # 1 on your homework, using your calculator!
Example
Do # 1 on your homework, using your calculator!

9. 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

10. 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.

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. 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 𝟒𝟏.𝟒𝟖
X = 92.6
𝒛= 𝒙−𝝁 𝝈
Do the next one on your own!

13. Guided Practice Worksheet Practice
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

14. 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?

15. 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 88% 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.

16. z = 1.23 z = .67 Example AP World x = 80 µ = 72 σ= 6.5
UBW x = 90 µ = 88 σ= 3
𝒛= 𝟖𝟎−𝟕𝟐 𝟔.𝟓
𝒛= 𝟗𝟎−𝟖𝟖 𝟑
z = 1.23
z = .67

17. 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

18. L = a and U = b
L = -1E99, U = x
L = x, U = 1E99

19. 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%

20. 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?

21. From Percentiles to Scores: z in Reverse
Sometimes we start with areas and need to find the corresponding z-score or even the original data value.
Example: What z-score represents the first quartile in a Normal model?

22. From Percentiles to Scores: z in Reverse (cont.)
Calculator method
2nd DISTR
invNorm(percent as a decimal)

23. In order to be in the top 40%, what score on the SAT did you have to have, given N(445, 55)?

24. What were the scores for the middle 50%?