# Chapter 3 Modeling Distributions of Data Essay

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3.1 Measuring Location in a Distribution Where do you Stand? p. 102 Introduce percentile and z­score

A) Measuring location: percentile
The
Pth Percentile of a distribution is the value with p percent of the observations less than or equal to it. (equal or below it) B) Measuring location: z­scores
1) Standardizing: Converting observations like for example, each height, from original values to standard deviation units from the mean
2) Standardized value is the z­score. z­score =
X ­ mean s Examples: Heights for Datoc’s Statistics Class Variable n Mean std. dev.
Min Q
M
Q
Max.
1
3
Height
25
67 4.29
60
63
66
69 75 1. Joe is 70 in. What is the z­score?

2. Mark is 63 in. What is the z­score?

3. Which person is farther away from the mean?

Review definitions for percentiles and z­score: 1.
An 8 year old boy who is 4’5″ (53 inches) tall is in the 86th percentile for height for his age. Explain what does this mean.

2.
Scores on a history test have average of 80 with standard deviation of 6.
a. What is the z ­score for a student who earned a 75 on the test? Interpret this number.
b. What is the z­score for a student who earned a 90 on the test? Interpret this number.
c. Which student is closer to the average score? Explain. In groups, go to this link: 1) http://academic.evergreen.edu/curricular/doingscience/flash/dice.html 2) click on other simulations involving dice 3) Change the number of dice to two. and click on roll multiple times. ● Be sure you can explain the difference between the shape of the red histogram and the shape of the blue histogram.
● Notice how much (or little) the average and standard deviation of the blue histogram change.
● Compare the predicted mean and standard deviation with the values you get from this simulation.
● What is the shape of the blue graph?

Density Curves­ A density curve is a curve that gives a rough description of a distribution. The curve is

smooth, so any small irregularities in the data are ignored.

Properties of a Density Curve:

a. Density curves are always drawn above the horizontal axis on a graph.
b. The area under a density curve (between the curve and the horizontal axis) is always defined as 1 unit.
c. The area under a density curve between two values is the proportion of observations in the data set that fall between those two values.
d. The median of a density curve is the equal­areas point, the point that divides the area under the curve in half.

e.

The mean of the density curve is the balance point, the point at which the curve

would balance if made of solid material.

Empirical (Normal) Rule Worksheet

Approximately 68% of the data values fall within 1 standard deviation of the mean. Approximately 95% of the data values fall within 2 standard deviation of the mean. Approximately 99.7% of the data values fall within 3 standard deviation of the mean.

1. Horse pregnancies: Bigger animals tend to carry their young longer before birth. The length of horse pregnancies from conception to birth varies according to a roughly normal distribution with mean 336 days and standard deviation of 3 days. Answer the following questions:
a. Almost all (99.7%) horse pregnancies fall in what range of lengths?

b. What percent of horse pregnancies are longer than 339 days?

2. Eggs: A truck is loaded with cartons of eggs that weigh an average of 2 pounds each with a standard deviation of 0.1 pound.
A histogram of these weights looks very much like a normal distribution. a. What percent of the cartons weigh less than 2.1 pounds? _____________ b. What percent of the cartons weigh less than 1.8
pounds?