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Lesson 24 Statistics

During this lesson we will focus on 4 statistical measures: 1) Measures of Central Tendency: mean, median, mode 2) Range 3) Stem-and-Leaf Plots 4) Box-and-Whisker Plots including how to do them on the TI-83 Calc

Measures of Central Tendency: There are three 1) Mean:

The sum of all the values in the data set divided by the number of values there are

2) Median:

The middle number of the data set when the values are placed in order. If there is no middle number, take the mean of the two middle numbers.

3) Mode:

The value that occurs most often in a data set. There may be more than one mode or no mode at all

Find the mean, median, and mode (in that order) of the salaries of the 10 employees of the Rhodes Company. Three workers earn $15,500 each, two earn $12,300 each, and five earn $17,050 each.

SOLUTION: Find the Mean: 15500+15500+15500+12300+12300+17050+ 17050+17050+17050+17050 10 Mean = 15,635 Find the Median:

Place numbers in order

12300, 12300, 15500, 15500, 15500, 17050, 17050, 17050, 17050, 17050 There is no middle number, so average the two middle numbers: 15500 + 17050 2 Median = 16,275 Find the Mode: The number that occurs more than any other is $17050 Mode = 17,050

The ages of a group of friends are: 35, 37, 44, 32, 46, 53, 27, 27. 3. Find the mean 4. Find the median 5. Find the mode

Mean:

35+37+44+32+46+53+27+27

=

37.625

8

Median:

Place numbers in order 27, 27, 32, 35, 37, 37, 46, 53 There is no middle number, so average the two middle #’s 35 + 37 = 36 2

Mode:

The number that occurs more than any other is 27

The Range of a set of data is the maximum value minus the minimum value: For the set of data in the previous problem, the range would be found by: 53 - 27 = 26

A Stem-and-Leaf Plot is simply a way of organizing a set of data. On the left side you put the common (tens and hundreds) digits of all the data values and on the right side go all the ones digits. We will put the data values from the previous Try This problem in a stem-and-leaf plot: 35, 37, 44, 32, 46, 53, 27, 27

2

7, 7

3

2, 5, 7

4

4, 6

5

3

The first row corresponds to the numbers 27 and 27, the second row is 32, 35, 37, the third row is 44 and 46 and the last row is 53

Identify the median in the stem-and-leaf plot below.

14

1, 5, 7

15

0, 2, 5, 6, 9

16

0, 4, 4, 8, 8, 9

17

1, 2, 3

Solution: 160

Solution: there are 17 numbers in the stem-and-leaf plot. So the 9th number will be the median (the number in the middle). If we count nine numbers down from 141 (the first value) we get 160. There should be 8 numbers to the left and right of 160.

Find the mean and median of each of the following stem-andleaf plots below.

A)

7

2, 3, 7

8 9

B)

30

4, 5, 5

4, 6, 9

31

3, 7, 8, 9

5, 8

32

1, 3, 5, 7

SOLUTION: a) Mean:

72+73+77+84+86+89+95+98

=

84.25

8 Median:

There are 8 values so take the mean of the middle two 84 + 86

=

85

2

b) Mean:

304+305+305+313+317+318+319+321+323+325+327 = 316.09 11

Median:

There are 11 values, so the 6th value is the middle one 304+305+305+313+317+318+319+321+323+325+327 318

Which data is represented by the stem-and-leaf plot below?

5

7, 8, 9

6

3, 5, 9

7

4, 5

SOLUTION: 57, 58, 59, 63, 65, 69, 74, 75

A Box-and-Whisker Plot is used to graphically compare a set of data. The box and whisker plot contains 5 values for a set of data. 1) Minimum

the smallest value in the data set

2) First Quartile

median of the first half of the data set

3) Median

the median of the entire data set

4) Third Quartile

median of the second half of the data set

5) Maximum

the largest value in the data set

The minimum and maximum values are the whiskers and the other three values are the boxes--as is shown on the next slides figure.

Box-and-Whisker Plot Min

Q1

Median

Q3

Max

The sizes of the boxes and whiskers can vary depending on the distribution of the data numbers. However the box and whisker plot is clearly divided into 4 regions: whisker, box, box, whisker. Each region contains 25% of the data numbers in it. This is a result of using medians which divide up the numbers evenly.

Draw a Box-and-Whisker plot for the following data: 37, 28, 34, 36, 29, 40, 29, 33, 30 Solution: First order the data set, you could even make a stem-and-leaf plot: 28 29 29 30 33 34 36 37 40 Let’s calculate the five values we need for our plot: Minimum:

the smallest value

28

Maximum:

the largest value

40

Median: 33

the 5th of the 9 numbers: 28 29 29 30 33 34 36 37 40

Q1:

the median of the lower half below the median of 33, there are four numbers so take the mean of the middle two: 29 and 29: 29

Q3:

the median of upper half : (36+37)/2

36.5

Now all we have to do is take our 5 values and make our box and whisker plot: Min: 28

Q1: 29

Med: 33

28 29

20

Q3: 36.5

33

30

36.5

Max: 40

40

40

Always remember to put a number line below your box and whisker plot before beginning to ensure that it is shaped accurately.

Make a Box and Whisker plot of the data below: 26, 37, 28, 34, 36, 29, 40, 29, 33, 30

First order the data set: 26, 28, 29, 29, 30, 33, 34, 36, 37, 40 Minimum

26

Maximum

40

Median mean of two middle numbers Q1

31.5

median of lower half, below median of 31.5.

Q3

Use 26 28 29 29 30

29

median of upper half, above median of 31.5

Use 33 34 36 37 40 26

20

29

31.5

30

36 36

40

40

Notice that when you are calculating the values for Q1 and Q3 you take the lower half of the data (all the numbers below the median) and the upper half of the data (all the numbers above the median). You do not actually include the median with either of the sets. However, if you had to average two numbers to get the median, you do use those two numbers. Our last topic will be instructions to create a box and whisker plot on your TI - 83 Graphing Calculator:

Making a Box-and-Whisker Plot on the Calculator: As we go through this use the values from the previous Try This problem: 26, 37, 28, 34, 36, 29, 40, 29, 33, 30 1) Go to the STAT menu and select edit 2) Make sure to clear out any values in the calculator 3) Enter all the values from the data set in L1, they do not have to be in order. 4) Press 2nd and Y= to go to the statplot menu 5) Make sure each plot: (Plot 1, Plot 2,… is off) if not arrow to the word On and press enter to turn it Off. 6) Go back to Plot 1 and press enter to go to that menu 7) Turn Plot 1 on by arrowing to the word Off and pressing Enter

8) There should be 6 graph pictures, arrow to the right until you get to the bottom middle one which is a box plot. 9) The rest of the choices should be set--don’t change them. 10) Press zoom and choose option 9 (STAT) 11) You should get a box plot of your data 12) If you press the Trace button and move back and forth, the graph should show the values for Min, Q1, Med, Q3, and Max.

Go back to the answer for the try this problem and check that the values agree. If they don’t check the original data you entered to make sure you didn’t make a mistake entering the numbers. Enjoy this convenience as you work your summary problems and assignments.

Summary Questions: Turn these in to your teacher and do not begin the assignment on the next slide until you receive a response on your summary questions. 1) How do you find the median of a set of data that has an odd number of values? An even number of values? 2) Give a set of data with: a) one mode b) no mode c) two modes 3) What are the 5 key values of a box and whisker plot? 4) Make a stem and leaf plot of the following set of data: 75, 76, 79, 80, 83, 84, 90, 92, 94, 98, 99 5) What are the mean, median, mode and range of the data in #4? 6) Create a box and whisker plot of the data in #4 showing the five key values.