20, 22, 23, 24, 24, 25, 25, 27, 35

Are there any outliers?

Draw a skeleton boxplot.

Draw a modified boxplot.

35max

263

24

5.221

20min

Q

med

Q

25.17

25.55.22

)5.3(5.11

Q

25.31

25.526

)5.3(5.13

Q

35Outlier

Chebyshev’s & The

Empirical Rule

Objective

Calculate values

using Chebyshev’s

Theorem and the

Empirical Rule.

Relevance

To be able to calculate

values with

symmetrical and non-

symmetrical

distributions.

Describing Data in terms of the

Standard Deviation.

Test Mean = 80

St. Dev. = 5

Chebyshev’s Rule

The percent of observations that are within

k standard deviations of the mean is at

least

2

1100 1 %

k

Facts about Chebyshev

Applicable to any data set – whether it is

symmetric or skewed.

Many times there are more than 75% - this

is a very conservative estimation.

Using Chebyshev

2

3

4

4.472

5

10

75.0

89.0

94.0

95.0

96.0

99.0

%75

%89

%94

%95

%96

%99

Use your calculators to fill in the list

Interpret using Chebyshev

Test Mean = 80

St. Dev. = 5

1. What percent are between 70 and 90?

2. What percent are between 60 and 100?

80

%75

%94

Collect wrist measurements

(in)

Create distribution

Find st. dev & mean.

What percent is within 1 deviation of mean

Practice Problems

1. Using Chebyshev, solve the following problem for a

distribution with a mean of 80 and a st. dev. of 10.

a. At least what percentage of values will fall between 60

and 100?

b. At least what percentage of values will fall between 65

and 95?

%752k

%561005.1

115.1

2 xk

Normal Distributions

These are special density curves.

They have the same overall shape Symmetric

Single-Peaked

Bell-Shaped

They are completely described by giving its mean () and its standard deviation ().

We abbreviate it N(,)

Normal Curves….

•Changing the mean without changing the standard

deviation simply moves the curve horizontally.

•The Standard deviation controls the spread of a

Normal Curve.

Standard Deviation

It’s the natural measure of spread for Normal distributions.

It can be located by eye on a Normal curve. It’s the point at which the curve changes from concave

down to concave up.

Why is the Normal Curve Important?

They are good descriptions for some real data

such as

Test scores like SAT, IQ

Repeated careful measurements of the same quantity

Characteristics of biological populations (height)

They are good approximations to the results of

many kinds of chance outcomes

They are used in many statistical inference

procedures.

Empirical Rule

Can only be used if the data can be

reasonably described by a normal curve.

Approximately

68% of the data is within 1 st. dev. of mean

95% of the data is within 2 st. dev. of mean

99.7% of data is within 3 st. dev. of mean

Empirical Rule

What percent do you think……

www.whfreeman.com/tps4e

Empirical Rule (68-95-99.7 Rule)

In the Normal distribution with mean ()

and standard deviation ():

Within 1 of ≈ 68% of the observations

Within 2 of ≈ 95% of the observations

Within 3 of ≈ 99.7% of the observations

The distribution of batting average (proportion of hits) for the 432

Major League Baseball players with at least 100 plate appearances

in the 2009 season is normally distributed defined N(0.261, 0.034).

Sketch a Normal density curve for this distribution of batting averages. Label the points that are 1, 2, and 3 standard deviations from the mean.

What percent of the batting averages are above 0.329?

What percent are between 0.227 and .295?

%5.2

159. 193. 227. 261. 295. 329. 363.

%68

Scores on the Wechsler adult Intelligence Scale (a

standard IQ test) for the 20 to 34 age group are

approximately Normally distributed. N(110, 25).

What percent are between 85 and 135?

What percent are below 185?

What percent are below 60?

35, 60, 85, 110, 135, 160, 185

%68

%85.999985.15.100 or

2.5%

2. A sample of the hourly wages of employees who work in

restaurants in a large city has a mean of $5.02 and a st.

dev. of $0.09.

a. Using Chebyshev’s, find the range in which at least

75% of the data will fall.

b. Using the Empirical rule, find the range in which at

least 68% of the data will fall.

devst.2%75 38.566.4

devst.1%68 11.593.4

The mean of a distribution is 50 and the standard deviation is

6. Using the empirical rule, find the percentage that will fall

between 38 and 62.

%95.2 devst

A sample of the labor costs per hour to assemble a certain

product has a mean of $2.60 and a standard deviation of

$0.15, using Chebyshev’s, find the values in which at least

88.89% of the data will lie.

05.315.23 sx

..3%89.88 devst

Homework

Worksheet

Quiz Friday 9/25 on Boxplots & outliers