Unit 3 Lesson 1(4.1)

Numerical Methods for Describing Data4.1: Describing the Center of a Data Set

60% of the sample was satisfied with their cell phone service.

6.0159ˆ p

What values are used to describe categorical data?Suppose that each person in a sample of 15 cell phone users is asked if he or she is satisfied with the cell phone service.

Here are the responses:Y N Y Y Y N N Y Y N Y Y Y N N

What would be the possible responses?

Find the sample proportion of the people who answered “yes”:

Pronounced p-hatThe population proportion is

denoted by the letter p.

Population Parameter -

• Fixed value about a population• Typically unknown

Suppose we want to know the MEAN length of all the fish in Lake Lewisville . . .

Is this a value that is known?

Can we find it out?

At any given point in time,

how many values are

there for the mean length of fish in the

lake?

Sample Statistic -• Value calculated from a

sample

Suppose we want to know the MEAN length of all the fish in Lake Lewisville.

What can we do to estimate this unknown population characteristic?

Measures of Central TendencyMedian - the middle value of the data;

it divides the observations in half

To find: list the observations in numerical order

even is if values middle two the of average

odd is is value middle single median sample

n

n

Where n = sample size

Suppose we catch a sample of 5 fish from the lake. The lengths of the fish (in inches) are listed below. Find the median length of fish.

3 4 5 8 10

The numbers are in order & n is odd – so

find the middle observation.

The median length of fish is 5 inches.

Suppose we caught a sample of 6 fish from the lake. The median length is …

3 4 5 6 8 10

The numbers are in order & n is even – so find the middle two observations.

The median length is 5.5

inches.

Now, average these two values.

5.5

Measures of Central TendencyMean is the arithmetic average.

–Use m to represent a population mean

–Use x to represent a sample mean

Formula:

S is the capital Greek letter sigma – it means to sum the values that

follow

Population Parameter

Sample Statistic

m is the lower case Greek letter mu

Suppose we caught a sample of 6 fish from the lake. Find the mean length of the fish.

3 4 5 6 8 10

To find the mean length of fish - add the observations and divide

by n.

61086543

What happens to the median & mean if the length of 10 inches was 15 inches?

3 4 5 6 8 15

The median is . . .

5.5

The mean is . . .

61586543

6.833

What happened?

What happens to the median & mean if the 15 inches was 20?

3 4 5 6 8 20

The median is . . .

5.5

The mean is . . .

62086542

7.667

What happened?

Some statistics that are not affected by extreme values . . .

Is the median affected by extreme values?

Is the mean affected by extreme values?

NO

YES

Suppose we caught a sample of 20 fish with the following lengths. Create a histogram for the lengths of fish. (Use a class width of 1.)

Mean =Median =

3 5 6 10 6 7 7 8 4 5 6 4 7 5 9 9 8 7 6 8

6.5

Calculate the mean and median.

6.5

Look at the placement of the mean and median in this symmetrical distribution.

Suppose we caught a sample of 20 fish with the following lengths. Create a histogram for the lengths of fish. (Use a class width 1.)

Mean =Median =5.5

6.8

Calculate the mean and median.Look at the placement of the

mean and median in this skewed distribution.

3 5 6 10 15 7 3 3 4 5 6 4 12 5 3 4 8 13 11 9

Suppose we caught a sample of 20 fish with the following lengths. Create a histogram for the lengths of fish. (Use a class width of 1.)

Mean =Median =

8.57.75

Calculate the mean and median.Look at the placement of the

mean and median in this skewed distribution.

3 5 6 10 10 7 10 8 9 5 6 4 9 10 9 9 10 7 10 8

Recap:• In a symmetrical distribution, the mean and

median are equal.• In a skewed distribution, the mean is pulled

in the direction of the skewness.

• In a symmetrical distribution, you should report the mean!

• In a skewed distribution, the median should be reported as the measure of center!

Summary

p = Population Proportion (parameter)

= Sample Proportion (statistic) = Population Mean (parameter) = Sample Mean (statistic)When describing center…

Measure of CenterSymmetric MeanSkewed Median

Homework

• Pg.110:

#4.1, 4.2ab, 4.3, 4.14