2'4: Measures of Center

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2.4 Measures of Center

• We wish to find a value at the center or middle
  of a data set.
• Mean
• Median
• Mode
• Midrange

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MEAN

• ARITHMETIC AVERAGE
• the number obtained by adding the values and
  dividing the total by the number of values

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Find the mean of the following set of 5 quiz
scores

• 8, 3, 5, 6, 10
• What score would be needed on a 6th quiz to bring
  the quiz average up to 7?

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Notation

• ? denotes the addition of a set of values
• x is the variable usually used to represent the
  individual data values
• n represents the number of data values in a
  sample

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Notation
is pronounced x-bar and denotes the mean of a
set of sample values
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Notation
is pronounced x-bar and denotes the mean of a
set of sample values

• µ is pronounced mu and denotes the mean of all
  values in a population

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MEDIAN

• the middle value when the original data
  values are arranged in order
• Find the median of the following set of data
• 9, 3, 4, 7, 5, 2, 7

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4, 8, 8, 9, 11, 13 no exact middle
-- shared by two numbers
(even number of values)
8 9
2
MEDIAN is 8.5

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MODE

• The score that occurs most frequently
• the only measure of central tendency that can be
  used with nominal data

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Find the mode
11
Find the mode
12
Find the mode
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MIDRANGE

• the value midway between the highest and lowest
  values in the data set

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MIDRANGE

• the value midway between the highest and
  lowest values in the data set

Midrange
high (max X) low (min X)
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Find the midrange

• 7, 10, 2, 8, 12, 6, 6, 9

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Round-off Rule for Statistics

• Never round off in the middle of a calculation!!!
• Carry one more decimal place than is present in
  the original set of values

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Find the mean, median, mode, and midrange for the
following sets of data

• 3, 4, 4, 5, 6, 6, 7, 8, 9
• 1, 3, 5, 7, 8, 9
• 4, 4, 4, 4, 4, 4, 4, 5, 5, 8

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• 3, 4, 4, 5, 6, 6, 7, 8, 9
• Mean 5.8, Median 6,
• Mode 4 and 6, Midrange 6
• 1, 3, 5, 7, 8, 9
• Mean 5.5, Median 6,
• No Mode, Midrange 5
• 4, 4, 4, 4, 4, 4, 4, 5, 5, 8
• Mean 4.6, Median 4,
• Mode 4, Midrange 6

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Finding Stats Using Technology

• P. 65 5
• Technology Notes p. 64
• Finding the mode on the TI-83/84
• Sort the data Stat - SortA( - Enter - Lx) -
  Enter
• Return to list and scroll down finding the
  largest group of data.

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Best Measure of Center

• Advantages - Disadvantages
• Page 63

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Mean from a frequency table using technology

• P. 67 20
• Enter midpoints in L1, frequency in L2
• Stat - Calc - 1VarStats L1, L2

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Mean of data in a Frequency Table
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Mean from a Frequency Table

• use class midpoint of classes for variable x

x class midpoint f frequency
? f n
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2079
67.1
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Definitions

• Symmetric
• Data is symmetric if the left half of its
  histogram is roughly a mirror of its right
  half.
• Skewed
• Data is skewed if it is not symmetric and if
  it extends more to one side than the other.

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Skewness (p.63)
Figure 2-11 (b)
Mode Mean Median
SYMMETRIC
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Skewness (p. 63)
Figure 2-11 (b)
Mode Mean Median
SYMMETRIC
Mean
Mode
Median
Figure 2-11 (a)
SKEWED LEFT (negatively)
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Skewness (p. 63)
Figure 2-11 (b)
Mode Mean Median
SYMMETRIC
Mean
Mean
Mode
Mode
Median
Median
Figure 2-11 (a)
SKEWED LEFT (negatively)
SKEWED RIGHT (positively)
Figure 2-11 (c)