Normal Distribution and Z-scores Intro to Statistics

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.

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)?

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. µ

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

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

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

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

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

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 = 22.45 ft.

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

How do I find the data point if I know the z-score? You have a Normal Distribution with mean µ = 235.7 and Standard σ=41.58. Which data point has a z-score of -3.45? −𝟑.𝟒𝟓= 𝒙−𝟐𝟑𝟓.𝟕 𝟒𝟏.𝟒𝟖 X = 92.6 𝒛= 𝒙−𝝁 𝝈 Do the next one on your own!

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

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?

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 6.5. 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.

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

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

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

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 last between 20,000 and 35,000 miles? A 94% B 81% C 68% D 47%

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?

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?

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

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

What were the scores for the middle 50%?