1. Normal Curve
10.3 Z-scores

2. Vocabulary:
Empirical rule - Provides an estimate of the spread of the data in a normal distribution given the mean and the standard deviation.
Z-scores - A statistical measurement of a scores relationship to the mean in a set of data.

3. Normal Curve Bell-shaped, symmetrical curve
Transition points between cupping upward & downward occur at m + s and m – s
As the standard deviation increases, the curve flattens & spreads
As the standard deviation decreases, the curve gets taller & thinner

4. Can ONLY be used with normal curves!
Empirical Rule
Approximately 68% of the observations are within 1s of m
Approximately 95% of the observations are within 2s of m
Approximately 99.7% of the observations are within 3s of m
Can ONLY be used with normal curves!

5. X-axis is standard deviations
Normal Curve
34%
34%
68%
47.5%
47.5%
95%
Take time to slowly click through slide.
Stress that the Empirical Rule can ONLY be used with the assumption that the distribution is normal (bell-shaped curve). Sixty-eight percent of the ordered data of a normal distribution lies within one standard deviation of the mean. Ninety-five percent of the ordered data of a normal distribution lies within two standard deviations of the mean. And, 99.7% of the ordered data of a normal distribution lies within 3 standard deviations of the mean. The normal distribution here, in this example shown, has a mean of 0 and standard deviation of 1.
49.85%
49.85%
99.7%
X-axis is standard deviations

6. The height of male students at CHS is approximately normally distributed with a mean of 71 inches and standard deviation of 2.5 inches.
a) What percent of the male students are shorter than 66 inches?
b) Taller than 73.5 inches?
c) Between 66 & 73.5 inches?
About 2.5%
About 16%
About 81.5%

7. z score Standardized score Creates the standard normal density curve
Z-score normalizes samples of data so that you can compare data variability on the same “level” (m = 0 & s = 1)

8. What do these z scores mean?
-2.3
1.8
6.1
-4.3
Note: z-score is number of deviations above or below mean
2.3 s below the mean
1.8 s above the mean
6.1 s above the mean
4.3 s below the mean

9. What is the z -score of a value of 27, given a set mean of 24, and a standard deviation of 2?

10. What is the z -score of a value of 104
What is the z -score of a value of 104.5, in a set with μ = 125 and σ = 6.2

11. Find the value represented by a z -score of 2
Find the value represented by a z -score of 2.403, given μ = 63 and σ = 4.25

12. Normalized Data
Normalizing data allows you to compare different types of data
For example: Duka is part of a pigmy colony and is 54 inches tall. The average female pigmy height is 50 inches with a σ of 1.5 in. Amy is 66 inches tall and the average height for her race is 60 inches with a σ of 3 in. Who is considered taller?