AP StatisticsMonday, 14 September 2015

• OBJECTIVE TSW identify measures of central tendancy.

• Get a TI-83/TI-84 (mine or yours).

• TESTS are graded.

• Next QUIZ: Means, Medians, Boxplots– Friday, 18 September 2015

• Next TEST: Displaying Data– Wednesday, 23 September 2015

Means & Medians

PARAMETER

•Fixed value about a population

•Typically unknown

STATISTIC

•Value calculated from a sample

MEASURES OF CENTRAL TENDENCY• Median - the middle of the

data; 50th percentile

• The observations must be in numerical order.

• The median is the middle value if n is odd.

• The median is the mean of the middle two values if n is even.

NOTE: n denotes the sample size

•Mean - the arithmetic average

•Use m (mu) to represent a population mean

•Use x (x-bar) to represent a sample mean

xx

nFormula:

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

follow

parameter

statistic

MEASURES OF CENTRAL TENDENCY

•Mode – the observation that occurs the most often

•There CAN be more than one mode.

•If all values occur only once – there is no mode.

•It is not used as often as mean & median.

MEASURES OF CENTRAL TENDENCY

Suppose we are interested in the number of lollipops that are bought at a certain store. A sample of 5 customers buys the following number of lollipops. Find the median.

2 3 4 8 12

The numbers are in order & n is odd – so

find the middle observation.

The median is 4 lollipops!

Suppose we have sample of 6 customers that buy the following number of lollipops. The median is …

2 3 4 6 8 12

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

observations.

The median is 5 lollipops!

Now, average these two values.

5

Suppose we have sample of 6 customers that buy the following number of lollipops. Find the mean.

2 3 4 6 8 126

2 3 4 6 8 12To find the mean number of

lollipops add the observations and divide by n.

5.833x

What would happen to the median & mean if the 12 lollipops were 20?

2 3 4 6 8 206

2 3 4 6 8 20

The median is . . .

5

The mean is . . .

7.17

What happened?

What would happen to the median & mean if the 20 lollipops were 50?

2 3 4 6 8 506

2 3 4 6 8 50

The median is . . .

5

The mean is . . .

12.17

What happened?

RESISTANT•Statistics that are not affected by outliers

•Is the median resistant?

►Is the mean resistant?

YES

NO

AP StatisticsTuesday, 15 September 2015

• OBJECTIVE TSW identify measures of central tendancy.

• Get a TI-83/TI-84 (mine or yours).

• CHANGE: Assignment due Tomorrow– WS Means and Medians

• Next QUIZ: Means, Medians, Variability– Friday, 18 September 2015

• Next TEST: Displaying Data– Wednesday, 23 September 2015

T-Shirts???

T-Sh

irts?

??

Now find how each observation deviates from the mean.

What is the sum of the deviations from the mean?

Look at the following data set. Find the mean.

22 23 24 25 25 26 29 30

25.5x

x x 0

Will this sum always equal zero?

YESThis is the deviation from

the mean.

Look at the following data set. Find the mean & median.

Mean =Median =

21 23 23 24 25 25 26 2626 27

27 27 27 28 30 30 30 31 3232

27

Create a histogram with the data (use an x-scale of 2). Then find the mean and

median.

27

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

Look at the following data set. Find the mean & median.

Mean =Median =

22 29 28 22 24 25 28 2125

23 24 23 26 36 38 62 23

25

Create a histogram with the data (use an x-scale of 8).

Then find the mean and median.

28.176

Look at the placement of the mean and median in this right-skewed (positively skewed) distribution.

Look at the following data set. Find the mean & median.

Mean =Median =

21 46 54 47 53 60 55 5560

56 58 58 58 58 62 63 64

58

Create a histogram with the data. Then find the

mean and median.

54.588

Look at the placement of the mean and median in this left-skewed (negatively skewed)

distribution.

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!

TRIMMED MEAN

To calculate a trimmed mean:

• Multiply the % to trim by n

• Truncate that many observations from BOTH ends of the distribution (when listed in order)

• Calculate the mean with the shortened data set

Find a 10% trimmed mean with the following data.

12 14 19 20 22 24 25 2626 35

14 19 20 22 24 25 26 268

22

n = 10, and 10%(10) = 1

So remove one observation from each side!

Assignment•WS Mean and Median•Due Wednesday, 16 September 2015.

• Next QUIZ: Means, Medians, Variability– Friday, 18 September 2015

• Next TEST: Displaying Data– Wednesday, 23 September 2015