Means & Means & MediansMedians

Chapter 4Chapter 4

Parameter -Parameter -

►Fixed value about Fixed value about a populationa population

►Typical unknownTypical unknown

Statistic -Statistic -

►Value Value calculatedcalculated from a samplefrom a sample

Measures of Central Measures of Central TendencyTendency

►Median - the middle of the data; Median - the middle of the data; 5050thth percentile percentile Observations must be in Observations must be in

numerical ordernumerical order Is the middle single value if Is the middle single value if nn is is

oddodd The average of the middle two The average of the middle two

values if values if nn is even is even

NOTE: n denotes the sample size

Measures of Central Measures of Central TendencyTendency

►Mean - the arithmetic averageMean - the arithmetic average Use Use to represent a population to represent a population

meanmean Use Use xx to represent a sample to represent a sample

meanmeanFormulaFormula: : is the capital Greek

letter sigma – it means to sum the values that

follow

parameter

statistic

Measures of Central Measures of Central TendencyTendency

►Mode – the observation that Mode – the observation that occurs the most oftenoccurs the most often Can be more than one modeCan be more than one mode If all values occur If all values occur onlyonly once – once –

there is no modethere is no mode Not used as often as mean & Not used as often as mean &

medianmedian

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

22 3 3 4 4 8 8 12 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 Suppose we have sample of 6 customers that buy the following number of lollipops. The median is …the following number of lollipops. The median is …

22 3 3 4 4 6 6 8 8 12 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 Suppose we have sample of 6 customers that buy the following number of lollipops. Find the mean.the following number of lollipops. Find the mean.

22 3 3 4 4 6 6 8 8 12 12

To find the mean number of lollipops add the observations

and divide by n.

61286432 833.5x

Using the calculator . . .Using the calculator . . .

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

22 3 3 4 4 6 6 8 8 20 20

The median is . . .

5

The mean is . . .

62086432

7.17

What happened?

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

22 3 3 4 4 6 6 8 8 50 50

The median is . . .

5

The mean is . . .

65086432

12.17

What happened?

Resistant -Resistant -

►Statistics that are not affected by Statistics that are not affected by outliersoutliers

►Is the median resistant?Is the median resistant?

►Is the mean resistant?Is the mean resistant?

YES

NO

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

5.25x

0

Will this sum always equal zero?

YESThis is the deviation from

the mean.

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

Mean =Mean =

Median =Median =

21 23 23 24 25 25 26 2626 27

27 27 27 28 30 30 30 3132 32

27Create a histogram with

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

Mean =Mean =

Median =Median =

22 29 28 22 24 25 2821 25

23 24 23 26 36 38 6223

25Create a histogram with

the data. (use 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 distribution.

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

Mean =Mean =

Median =Median =

21 46 54 47 53 60 55 5560

56 58 58 58 58 62 63 64

58Create a histogram with

the data. Then find the mean and median.

54.588

Look at the placement of the mean and

median in this skewed left distribution.

Recap:Recap:►In a symmetrical distribution, the In a symmetrical distribution, the

mean and median are equal.mean and median are equal.►In a skewed distribution, the mean is In a skewed distribution, the mean is

pulled in the direction of the skewness. pulled in the direction of the skewness. “The tail pulls the mean.”“The tail pulls the mean.”

►In a In a symmetrical symmetrical distribution, you distribution, you should report the should report the meanmean..

►In a In a skewedskewed distribution, the distribution, the medianmedian should be reported as the measure of should be reported as the measure of center.center.

Trimmed mean:Trimmed mean:To calculate a trimmed mean:To calculate a trimmed mean:►Multiply the % to trim by Multiply the % to trim by nn►Truncate that many observations Truncate that many observations

from from BOTHBOTH ends of the ends of the distribution (when listed in order)distribution (when listed in order)

►Calculate the mean with the Calculate the mean with the shortened data setshortened data set

Find a 10% trimmed mean with the following Find a 10% trimmed mean with the following data.data.

1212 14 19 14 19 20 20 22 22 2424 2525 2626 2626 2929

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

Therefore, remove one observation from each side!

228

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Homework:Homework:

Page 27, “Mean and Median WS”Page 27, “Mean and Median WS”