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Measures of Central Tendency Mode, Median, Mean, Range (Range, Frequency Distribution)
Measures of Central Tendency Mode, Median, Mean, Range
(Range, Frequency Distribution)
MODE
• 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 ?
• 3
• The mean is ?
• $159,000 3
• $53,000
Range
• 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?
1. 90
2. 90
3. 90
4. 90
1. 96
2. 90
3. 89
4. ??
360 275 _
= 85
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
4. 3
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
Lower
extreme
Upper
extreme
Lower
quartile
Upper
quartile
Median (middle
quartile)
Friday, August 30, 2013
Example #2 • Even Data Set:
152 153 153 154 156 158 160 161 163 163 164 164
Lower
extreme
Upper
extreme
159
Median
Middle (2nd)
quartile
Lower
(1st)
quartile
153.5
Upper
(3rd)
quartile
163
Friday, August 30, 2013
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
Friday, August 30, 2013
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
Friday, August 30, 2013
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
6 8
7 6 6 7 8
8 1 1 1 9
Using a Stem-and-Leaf
Plot Stem Leaf
5 0 3 6 6
6 8
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
Friday, August 30, 2013
Stem & Leaf Plot
Stem Leaf
6 4
7 3 8 9
8 1 3 5 5 5
9 3 5 7 7
10 0
Mode = 85
Range = 100 – 64
= 36
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 =
85
85.4
Problem 2 – Stem-Leaf
Stem Leaf
0 6 9 9
1 1 1 2 2 2 3 5 5 5 5 7 8
2 0 0
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
5. Interquartile Range
44
39
49.5 15
10.5
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
Friday, August 30, 2013
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
5. Interquartile Range
6. #2
#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 _
= 85
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
6 8
7 6 6 7 8
8 1 1 1 9
1.
2. Mean= 76.3
Median=76
Mode=81
Range=39
3. Extremes= none
Quartiles= 10, 38
4.
17 22 27 32 37 42 47 52 57
Friday, August 30, 2013
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
