Measures of Central Tendencies

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Measures of Central Tendencies

• Week 2

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Definitions

• Mean or averageThe sum of a set of data divided
  by the number of data.(Do not round your answer
  unless directed to do so.)
• Median The middle value, or the mean of the
  middle two values, when the data is arranged in
  numerical order. 
• Mode The value (number) that appears the most
  often.It is possible to have more than one mode
  (bimodal), and it is possible to have no mode. 
  If there is no mode, write "no mode", do not
  write zero (0)
• Range the difference of the highest and lowest
  value
• Midrange sum of highest and lowest values,
  divided by two.

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MEAN

• Advantages     Most popular measure in fields
  such as business, engineering and computer
  science.     It is unique - there is only one
  answer.     Useful when comparing sets of data.
  Disadvantages     Affected by extreme values
  (outliers)

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Mean formula
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MEDIAN

• Advantages     Extreme values (outliers) do
  not affect the median as strongly as they do the
  mean.     Useful when comparing sets of
  data.     It is unique - there is only one
  answer.Disadvantages     Not as popular as
  mean.

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MODE

• Advantages
• Extreme values (outliers) do not affect the mode.
• Can be used for non-numerical data colors, ect.
• Disadvantages
• Not as popular as mean and median
• Not necessarily unique - may be more than one
  answer
• When no values repeat in the data set, the mode
  is every value and is useless.
• When there is more than one mode, it is difficult
  to interpret and/or compare.

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What happens if.

• If we replace the lowest grade with a zero

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Distribution of data

• Skewed left, negatively skewed mediangtmean
• b) Skewed right, positively skewed meangtmedian
• c) Symmetric, meanmedianmode

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Mean        Median     5, 5, 10, 10, 10,
15, 15, 15, 20, 25           Listing the data in
order is the easiest way to find the median.
         The numbers 10 and 15 both fall in the
middle.  Average these two numbers to
get the median  (1015)/2 12.5    
                                                  
                       Mode    Two numbers
appear most often  10 and 15.                 
There are three 10's and three 15's.             
     In this example there are two answers for
the mode. Range The difference of highest
lowest 25 5 20 Midrange Average of
highest lowest (525)/2 15
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EX. 2 - On his first 5 biology tests, Bob
received the following scores  72, 86, 92, 63,
and 77.  What test score must Bob earn on his
sixth test so that his average (mean score) for
all six tests will be exactly 80?  Show how you
arrived at your answer.
Possible solution method          Set up an
equation to represent the situation. 
Remember to use all 6 test scores               
                                 72 86 92
63 77 x     80                              
                                  6
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EX. 3 Grouped Data On a statistics
examination, 7 students received scores of 95, 9
students received 90, 6 students received 85, 4
students received 80, there was one 75, 3
students with a 70, and one student received 65.
The mean score on this examination was (nearest
10th)
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EX. 4 Grouped Data Using this frequency table
of test scores Find the Mean, Median, Mode,
Range, and Midrange
Grade Frequency
100 1
95 3
90 5
85 8
80 7
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Box and Whisker Plots

• Determine the five key values
• (5 number summary)
• Minimum value
• Maximum value
• Median (2nd Quartile) or Q2
• 1st Quartile median of values less than Q2
• 3rd Quartile - median of values greater than Q2
• - Interquartile range Q3-Q1

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Ex. Construct a box-and-whisker plot for the
following dataThe data  Math test scores 80,
75, 90, 95, 65, 65, 80, 85, 70, 100
1) Write the data in numerical order. Find the
five key values.  median (2nd quartile)
80first quartile 70third quartile
90minimum 65maximum 100
2) Create a number line and plot values below it
3) Draw a box with ends through the points for
the first and third quartiles.  Then draw a
vertical line through the box at the median
point.  Now, draw the whiskers (or lines) from
each end of the box to these minimum and maximum
values.
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Outlier one value that is an extreme value
You may see a box-and-whisker plot, like the one
below, which contains an asterisk.
10 is an extreme value and is placed on the
graph but not part of the box or whisker
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25 of data
25 of data
25 of data
25 of data
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