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. 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

9. 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.

10. # 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?

11. 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!

12. Guided Practice Worksheet Practice
The distribution of weights of 9-ounce bags of a particular brand of potato chips is approximately Normal with mean of 9.12 ounces and standard deviation of 0.05 ounces. 1. Label the model

13. What would be the z- score for a bag weighing 9.25 z?
What would be the z-score for a bag weighing 9.05 oz?
Which bag weight is more unlikely?

14. 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.

15. 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

16. How likely is it that a bag weighs less than 9 oz?
What is the probability that a bag weighs between 9 and 9.1 oz?
What is the probability that a bag weighs more than 9.2 oz?

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. How likely is it that a bag weighs less than 9 oz?
What is the probability that a bag weighs between 9 and 9.1 oz?
What is the probability that a bag weighs more than 9.2 oz?

20. What are Percentiles?
The percentile of a distribution is the percent of observations less than it.
What is the percentile of a bag weighing 9.17 oz?
What is the percentile of a bag weighing 9.15 oz?

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%?