Measures of Central Tendency

1
Measures of Central Tendency

• Beyond the Three Ms

2
Review Mean

• Add up the data values
• Divide the total by the number of data

3
Review Median

• Sort the data into size order
• Find the middle one or middle pair
• The median is the middle value
• or the mean of the middle two

4
Review Mode

• Make a Frequency Table of your data
• The mode is the value with the highest frequency
• The Modal Class is the interval with the highest
  frequency in grouped data

5
Sigma Notation

• Sigma S is the Greek letter S
• Sigma S means add up
• Think Sigma S S Sum Add

6
Sigma Notation for Mean

• The Mean of n lots of x values is
• Say x bar equals sigma x over n

7
The Moving Average

• Consider a 4-point moving average
• Find the mean of the first four values(a, b, c,
  d)
• Move along one
• Find the mean of the next four values(b, c, d,
  e)
• Move along one (c, d, e, f) etc.

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Moving Average example

• Finding a five point moving average

9
Moving Average example

• Finding a five point moving average

10
Moving Average example

• Finding a five point moving average

11
Moving Average example

• Finding a five point moving average

12
Moving Average example

• Finding a five point moving average

13
Moving Average example

• Finding a five point moving average

14
Moving Average example

• Finding a five point moving average

15
Moving Average example

• Finding a five point moving average

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Moving Average example

• Finding a five point moving average

17
Moving Average - Why?

• Moving average helps to iron out fluctuations
  in the data
• Can be used to identify overall trends
• Very useful for time series data e.g. monthly
  sales figures
• When graphing, plot the MA against the middle
  value used

18
Weighted Mean

• Your overall percentage for Statistics is made up
  as follows
• 50 Homeworks
• 30 Tests
• 20 Mocks
• We use a Weighted Mean to calculate your overall
  percentage.

19
Weighted Mean

• Write the percentages as decimals
• 50 Homeworks 0.5
• 30 Tests 0.3
• 20 Mocks 0.2
• Multiply each score by its weight then add them up

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Weighted Mean - example

• Homeworks 64 weighting 0.5Tests 59 weighting
  0.3Mock 78 weighting 0.2
• Overall Percentage 64 x 0.5 59 x 0.3 78 x
  0.2 65.3

21
Geometric Mean

• Instead of adding you multiply the data together
• Instead of dividing by the number of data (n) you
  find the nth root
• Use this to find the average of two or more
  multiplicative factors e.g. interest rates,
  percentage change

22
Geometric Mean - example

• A plant grows in height 20 in the first year,
  12 in the second year and 7 in the third year.
  What is the average percentage increase per year?
• Find the geometric mean of 1.2, 1.12 and 1.07
• 1.2 x 1.12 x 1.07 1.43808
• Cube root gives 1.1287 12.9

23
Summary of Means

• Arithmetic Mean (the normal one)
• Moving Mean (used to show overall trends in
  seasonal data)
• Weighted Mean (used when some values are more
  important than others)
• Geometric Mean (used to find the average of
  multiplicative changes)

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Data Transformations

• What is the effect on the mode, median and mean
  if all your data is
• Multiplied by 5
• Halved
• Increased by 100
• Decreased by a fifth
• Every (X) changed to (2X 5) ?

25
Data Transformations

• The effects are as follows
• Multiplied by 5
• Halved
• Increased by 100
• Decreased by a fifth
• Every (X) changed to (2X 5).

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Making Life Easier!

• Example Find the mean of 2003, 2009, 2004,
  1995, 2000
• Method Subtract 2000 from each value3, 9, 4,
  -5, 0
• Find the mean of these
• (394-50) / 5 2.2
• Add the 2000 back again 2002.2

27
Why use Scaling?

• Scaling means changing each data value x to some
  other value Ax B
• This is useful when we want to scale the data so
  that the mean is more useful e.g. scale test
  scores to a mean of 50
• Adding / Subtracting only affect the Ms
• Multiplying / Dividing affect the range as well
  (including IQR)

28
Summary

• Review simple average Three Ms
• Extend ideas about Mean
• Consider Data Transformations (Scaling)
• How might you use these techniques to analyse the
  data in your project?