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Measures of Central Tendency Mode, Median, Mean, Range

(Range, Frequency Distribution)

• The number that occurs most frequently in a group of numbers (data).

• 12, 18, 17, 10, 15, 23, 12, 9

• The mode is 12

• 9, 0, 1, 9, 5, 4, 9, 2

• The mode is 9

• 10, 5, 8, 10, 9, 8, 3, 6

• Both 10 and 8 are the modes.

Median

• The middle number in a set of data that is

arranged in either ascending or descending order.

• 12, 18, 17, 10, 15, 23, 12, 23, 9. Arrange in

ascending order.

• 9, 10, 12, 12, 15, 17, 18, 23, 23.

• Arrange in descending order.

• 23, 23, 18, 17, 15, 12, 12, 10, 9.

Median continued

• 9, 10, 12, 12, 15, 17, 18, 23, 23.

What is the middle number?

• 9, 10, 12, 12, 15, 17, 18, 23, 23

• 22, 8, 15, 60, 3, 7, 9, 9, 1, 2, 4.

Order the list of numbers

• 1, 2, 3, 4, 7, 8, 9, 9, 15, 22, 60.

What is the median?

• 1, 2, 3, 4, 7, 8, 9, 9, 15, 22, 60.

Median and Even Numbered Data Sets • What if you have TWO middle

numbers (even set)?

• 12, 18, 17, 10, 15, 23, 12, 23, 9, 7

• Order the numbers

• 23, 23, 18, 17, 15, 12, 12, 10, 9, 7 What is the middle number?

• 23, 23, 18, 17, 15, 12, 12, 10, 9, 7 – which is the median?

• You must find the mean or average of the two middle numbers by adding them and dividing by two.

• The median is 27 2 = 13.5

Median continued

• Find the median by dividing the total – 27, by the number of items added – 2.

• The median is 27 2

• 23, 23, 18, 17, 15, 12, 12, 10, 9, 7 – What is the median?

• 13.5

The mean, or arithmetic average, is found by adding a group of numbers and dividing the total by the number of items added.

Mean (Average)

• Find the mean of the following data: $48,000, $64,000, $47,000.

• The sum is ?

• $159,000

• The number of items added is ?

• The mean is ?

• $159,000 3

• $53,000

• The difference between the highest and lowest numbers in a set of data.

• $48,000, $64,000, $47,000

• Which is the highest number?

• $64,000

• Which is the lowest number?

• $47,000

• What is the range?

• $64,000 - $47,000 = $17,000

Range continued

• 1023, 456, 98, 401, 1700, -27

• Which is the highest number?

• 1700

• Which is the lowest number?

• -27

• What is the range?

• 1700 – (-27)

• =1700 + 27

• 1727

Finding the Missing Data June wants a 90% average in math class. So far

she has three scores of 96%, 90%, and 89%. What score does she need on her next test?

360 275 _

Another Example • Zena needs to ship four packages

with a total combined weight of 48

ounces. What is the average

weight of each package?

• The total is already provided, and

there are four objects.

48 4 = 12 ounces

Homework

Red Book: 247-251; 1-3 every page

Green: 217-220,224; # 1,5,10

Pg. 185, 187, 188; 1-4 on all

three pages

• Please take a seat and have a sheet of

paper ready with your name on it.

1. Did you remember the Quiz?

2. Define Median

3. 1,2,3,4,5,6 Median?

4. 4,2,5,3,1

1. Yes, No, Maybe….

2. Order, Middle

3. 3.5

Warm-up (use pencil please) A

2,4,3,11,8,7,5,3,2 1. Mean=

2. Median=

3. Mode=

4. Range=

5. Explain the two cases

for finding a median.

Warm-up B (use pencil please)

3,4,10,8,7,3,7,8,1 1. Mean=

2. Median=

3. Mode=

4. Range=

5. Explain when is there

no mode?

Warm-up C (use pencil please)

2,4,10,8,7,3,7,8,1,2 1. Mean=

2. Median=

3. Mode=

4. Range=

How to construct a Box & Whisker Plot

Box-and-Whisker Plots Creating a Box-and-Whisker Plot.

1. Place order in ascending data (small to big)

40, 19, 17, 58, 42, 52, 27, 37

17, 19, 27, 37, 40, 42, 52, 58

2. Determine the median

(37 + 40)/2 = 38.5

3. Find the median of the upper half of the data (upper quartile).

17, 19, 27, 37, 40, 42, 52, 58

(42 + 52)/2 = 47

4. Find the median of the lower half of the data (lower quartile).

17, 19, 27, 37, 40, 42, 52, 58

(19 + 27)/2 = 23

1, 2, 5, 6, 7, 9, 10, 12

Make a Box & Whisker Plot “B”

5, 7, 1, 6, 2, 12, 10, 9

1. Median

2. Lower Quartile

3. Upper Quartile

4. Range

5. Interquartile Range

Box-and-Whisker Plot

1, 2, 5, 6, 7, 9, 10, 12

1, 2, 5, 6, 7, 9, 10, 12: (6+7)/2 = 6.5 = median

1, 2, 5, 6: (2 + 5)/2 = 3.5 = lower quartile

7, 9, 10, 12: (9 + 10)/2 = 9.5 = upper quartile

1) Order the numbers from smallest to biggest

2) Find the middle number

3) Divide each new section in half and find the

middle number

3) Plot each value on a number line and make

the boxes

Friday, August 30, 2013

Example

• 45 9 27 15 34 30 42 7 11 • Place in order:

7 9 11 15 27 30 34 42 45 – Determine quartiles and extremes:

7 9 11 15 27 30 34 42 45

extreme

quartile

Median (middle

quartile)

Example #2 • Even Data Set:

152 153 153 154 156 158 160 161 163 163 164 164

extreme

Median

Middle (2nd)

quartile

Make a Box & Whisker

59, 61, 63, 65, 66, 66, 67, 67, 69, 70, 70: 66 = median

59, 61, 63, 65, 66, 66: (63 + 65)/2 = 64 = lower quartile

66, 67, 67, 69, 70, 70: (67 + 69)/2 = 68 =upper quartile

Problem 3

Box & Whisker

1, 1.5, 1.7, 2, 6.1, 6.2, 7: 2 = median

1, 1.5, 1.7, 2: (1.5 + 1.7)/2 = 1.6 = upper quartile

2, 6.1, 6.2, 7: (6.1 + 6.2)/2 = 6.15 = lower quartile

0 1 2 3 4 5 6 7 8

Stem & Leaf Plots Values are separated into a “stem” and a “leaf”. The leaf is

the last digit of the number. The stem are any numbers to

the left of the last digit.

256 Which digit is the leaf? 6

What is the stem? 25

Constructing a Stem-and-Leaf Plot Construct a stem-leaf plot using the following data:

56, 89, 53, 81, 76, 68, 77, 81, 76, 56, 78, 81, 50

Stem Leaf

5 0 3 6 6

7 6 6 7 8

8 1 1 1 9

Using a Stem-and-Leaf

Plot Stem Leaf

5 0 3 6 6

7 6 6 7 8

8 1 1 1 9

What is the mode for this set of

data? 81

What is the median? 76

What is the range? 39

What is the mean?

50 + 53 + (56 x 2) + 68 + (76 x 2) + 77 +

78 + (81 x 3) + 89 = 992

992 13 = 76.3

Stem & Leaf Plot

Stem Leaf

7 3 8 9

8 1 3 5 5 5

9 3 5 7 7

Mode = 85

Range = 100 – 64

Median = the mean of

the 7th and 8th

numbers. (85 + 85)/2

Mean = {64 + 73 + 78 + 79 + 81 + 83 + (85 x 3)

+ 93 + 95 + (97 x 2) + 100}/14 =

Problem 2 – Stem-Leaf

Stem Leaf

0 6 9 9

1 1 1 2 2 2 3 5 5 5 5 7 8

Mode = 15 Range = 20 – 6 = 14 Median = 13

Mean = {6 + (9 x 2) + (2 x 11) + (3 x 12) + 13 +

(4 x 15) + 17 + 18 + (2 x 20)}/17 = 13.5

Make a stem & leaf plot and a box & whisker

35, 36, 38, 40, 42, 42, 44, 47, 48, 49, 50, 50, 50

1. Median

2. Lower Quartile

3. Upper Quartile

4. Range

49.5 15

Warm-up Make a Box & Whisker A

59, 61, 63, 65, 66, 66, 67, 67, 69, 70, 70: 66 = median

59, 61, 63, 65, 66: = 63 = lower quartile

67, 67, 69, 70, 70: = 69 =upper quartile

58 60 62 64 66 68 70 72

66, 70, 67, 65, 66, 59, 63, 67, 69, 61, 70

2. June wants a 90% average in math class. So far she has three

scores of 96%, 90%, and 89%. What score does she need on her

next test? 1. Median

2. Lower Quartile

3. Upper Quartile

4. Range

#2 Finding the Missing Data June wants a 90% average in math class. So far

she has three scores of 96%, 90%, and 89%. What score does she need on her next test?

1. 90 2. 90 3. 90 4. 90

1. 96 2. 90 3. 89 4. ??

360 275 _

1. Construct a stem-leaf plot using the following data: 56, 89, 53, 81, 76, 68, 77, 81, 76, 56, 78, 81, 50

2. Find the mean, mode, median, and range for # 1.

3. Find the quartiles and extremes:

45, 9, 27, 15, 34, 30 , 42, 7, 11

4. Draw a box & whisker plot:

40, 19, 17, 58, 42, 52, 27, 37

Central Tendency

Quiz Key Stem Leaf

5 0 3 6 6

7 6 6 7 8

8 1 1 1 9

2. Mean= 76.3

Median=76

Mode=81

Range=39

3. Extremes= none

Quartiles= 10, 38

17 22 27 32 37 42 47 52 57

Quartiles & Extremes • In statistics, data is separated into four

equal parts called quartiles (think quarter which is 1/4th of a dollar).

• The median of the ENTIRE data set is in the middle and separates the data into two halves.

• The median of the UPPER half is the upper quartile.

• The median of the BOTTOM half is the lower quartile.

Quartiles & Extremes cont. The extremes are the highest and lowest

numbers.

The lowest value is called the lower extreme.

The highest value is called the upper extreme.

The data set MUST BE placed in order!!

Homework Red Book 253-254; 1-6

255-256; 1-2

Yellow Book 192; 1-5

193; 1,2,5,7 194; 1-2

Mode, Median, Mean, Range - myCCSDmy.ccsd.net/userdocs/documents/w5doHdkk6RvPgvR6.pdf · Mode,...

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