Section 24 Measures of Central Tendency

1
Section 2-4Measures of Central Tendency

• The measures of Center are mean, median mode and
  midrange.
• Suggested HW
• PP69-73 1-4 9-12 1718 for sect. 2-4
• Pp87-92 1-4 9-14 1718, 21-24 for sect2.5
• Pp99-101 1-10 13-24

2
Measures of Center

• 1) Mean Its the most important value used to
  describe center. It is what most people usually
  call average. Mean ?x/n
• x(bar) ?x/n (mean of a sample)
• µ ?x/N (Mean of all values in a population)
• Note Mean is sometimes affected by outliers
• 2) Median Its the middle value after putting
  them in order. If the number of values is even,
  add the two middle values divided by 2.
• Ex pp60-62

3
Measures of center

• 3) Mode It is the most repeating value or
  values. The set could be bimodal (2 modes),
  multimodal ( many modes). Sometimes, there may be
  no modes if no numbers or values are repeating.
• 4) Midrange It is the value midway between the
  highest and the lowest. MR(HVLV)/2
• Examples for mean, mode, median and midrange, go
  to pp60-64

4
Mean from a frequency distribution and weighted
mean

• Please read examples p66 for
• Mean from a frequency distribution
• X(bar) ?(fx) ?f
• x is the class midpoint
• b) Weighted mean ?(wx) ?w
• Exercises 17- 20 can help you understand these
  two mean values.

5
Which measure of center is the best?

• No single best answer for this question. Please
  refer to page 67 table 2-10 for a better
  understanding of the four measures of center.

6
Skewness

• If the left side of a data set is not roughly the
  mirror image of the right side, the data is said
  to be skewed in its distribution.
• Example p68

7
Measures of Variationsection 2-5

• Please read this section completely. Its one of
  the most important in this book. You dont need
  to memorize the formulas neither do the
  calculations. When you read, look for the way
  they interpret the results. This is what we will
  most likely do and this is what they really need
  from you.
• Lets get started with the example of single line
  system (4,7,7mins. of waiting times) and multiple
  line system (1, 3, 14 mins.) used by certain
  commercial banks (p 74). Think about- Which
  system makes customers happier? Why?

8
Measures of Variation

• Range Highest value Lowest value
• Standard deviation Its a measure of variation
  of values about the mean.
• S v(?(x xbar)2n-1) for a sample
• S vn?(x2) (?x)2 n(n-1) shortcut formula for
  a sample
• Procedures and examples pp7677
• s v?(x µ) N for a population
• The v should cover all symbols

9
Help in interpreting your results!

• Understand that the standard variation is a
  variation of all values from the mean
• It is usually a positive number meaning that
  larger values are signs of more variation. Its
  zero only when all the data values are the same.
  Note also that an outlier can increase the value
  of the standard deviation so be careful in
  interpreting such data.
• Now, PP76 77 as much as possible try to use
  table 2-11 p77

10
Variation

• Its always the square of the standard deviation
• S2 for variance of a sample
• s2 for variance of a population
• Variance is expressed in square units
• We will do more work with it in section 8-5
  and chapter 11.

11
Comparing variance in different populations

• To compare variation in different populations, we
  usually use the coefficient of variation because
  it has no units.
• CV sxbar100 for samples
• CV sµ100 for populations
• Higher percentages suggest more variation
• Ex p80

12
Standard Deviation from a Frequency Distribution

• S vn?(fx2) ?(fx)2n(n-1)
• x is the class midpoint
• v should cover all the symbols
• Examples Table 2-12 P81

13
How to interpret standard deviations pp81-86

• It measures the variation among values.
• Higher value ? bigger differences between the
  data values in the set
• 2) 95 of the data usually fall within 2 standard
  deviations of the mean
• For the sake of simplicity, let sacrifice
  accuracy
• S range/4
• Minimum Value Mean 2S
• Maximum Value Mean 2S
• Chebyshevs Theorem Percent or fraction of data
  within k standard deviation 1-1/k2, k?0
• Empirical Rule will be done in chapter 5 but
  for now, remember 68-95-99.7 for Bell-shaped
  distribution. p83

14
Section 2.6Measures of Relative Standing

• Measures of Relative Standing help us compare
  values either inside a set or not.
• Z scores give the position of a value x when
  comparing to the mean (comparing values from
  different data sets)
• Z (x xbar) S or Z (x-µ) s
• This concept will be used in chapter 5 Ex p
  93
• If the zscore lt -2 or gt 2 or the x value is
  below or above 2 standard deviations of the mean,
  the x value is considered as unusual. P94
• This will be very helpful ch 7 with hypothesis
  testing.

15
Measures of Relative Standing

• 2) Quartiles and Percentiles (comparing values
  within the same set)
• Quartiles divide the data set into 4 equal parts
• Q2 median or 50 of a sorted set
• Q1 (first half)/2 25 of the bottom part
• Q3 (second half)/2 75 of the bottom part
• Percentiles divide the data into 100 groups1
  each
• Percentile of value x
• (N. of values less than x)(T. N. of values) 100
  Ex pp95-97

16
Other terms associated with Quartiles or
Percentiles

• Interquartile range (IQR) Q3-Q1
• Semi-interquartile range (Q3-Q1)/2
• Midquartile (Q3 Q1)/2
• 10-90 percentile range p90-p10

17
Exploratory Data Analysis

• So far, youve seen the basic tools for
  describing, exploring and comparing data. Now,
  lets explore them.
• Exploratory data analysis Investigate data sets
  in order to find their characteristics. To do so,
  we need to use tools such as graphs, measures of
  center, measures of variation. Also you need to
  understand that outliers could be a mistype
  pp102103. Also see types of bell-shaped p104 and
  read pp106-108.

18
Some of the answersReview for test1

• True 2) stratified 3) Not random 4) Same, 5)
  40.4, 6) 4 yes, 7)0.90 8)65, 9)55.4, 10)4.8, 11)
  1.075, 12)69.8 13)0.9, 14)approximately 50, 15)
  answers may vary 16) at least 88, 17)5.7,
  18)2.8, 19) Ratio, 20) at least 86, 21)1.51,
  22)68, 23)prospective, 24)35, 25)Restaurant A
  57 493.98 22.23 Restaurant B 77 727.98 26.98