Understanding Basic Statistics Third Edition By Brase and Brase Prepared by: Lynn Smith Gloucester C

1
Understanding Basic StatisticsThird EditionBy
Brase and BrasePrepared by Lynn
SmithGloucester County College

• Chapter Three
• Averages and Variation

2
Measures of Central Tendency

• Mode
• Median
• Mean

3
The Mode

• the value that occurs most frequently in a data
  set

4
Find the mode

• 6, 7, 2, 3, 4, 6, 2, 6
• The mode is 6.

5
Find the mode

• 6, 7, 2, 3, 4, 5, 9, 8
• There is no mode for this data.

6
The Median

• the central value of an ordered distribution

7
To find the median of raw data

• Order the data from smallest to largest.
• Pick the middle value.
• or
• Compute the average of the middle two values

8
Find the median

• Data 5, 2, 7, 1, 4, 3, 2
• Rearrange 1, 2, 2, 3, 4, 5, 7

The median is 3.
9
Find the median
Data 31, 57, 12, 22, 43, 50 Rearrange 12, 22,
31, 43, 50, 57
The median is the average of the middle two
values
10
The Mean

• The mean of a collection of data is found by
• summing all the entries
• dividing by the number of entries

11
Find the mean
6, 7, 2, 3, 4, 5, 2, 8
12
Sigma Notation

• The symbol S means sum the following.
• S is the Greek letter (capital) sigma.

13
Notations for mean

• Sample mean

• Population mean
• Greek letter (mu)

14
Number of entries in a set of data

• If the data represents a sample, the number of
  entries n.
• If the data represents an entire population, the
  number of entries N.

15
Sample mean
16
Population mean
17
Resistant Measure

• a measure that is not influenced by extremely
  high or low data values

18
Which is less resistant?

• Mean
• Median

• The mean is less resistant. It can be made
  arbitrarily large by increasing the size of one
  value.

19
Trimmed Mean

• a measure of center that is more resistant than
  the mean but is still sensitive to specific data
  values

20
To calculate a (5 or 10) trimmed mean

• Order the data from smallest to largest.
• Delete the bottom 5 or 10 of the data.
• Delete the same percent from the top of the data.
• Compute the mean of the remaining 80 or 90 of
  the data.

21
Compute a 10 trimmed mean

• 15, 17, 18, 20, 20, 25, 30, 32, 36, 60

• Delete the top and bottom 10
• New data list
• 17, 18, 20, 20, 25, 30, 32, 36
• 10 trimmed mean

22
Measures of Variation

• Range
• Standard Deviation
• Variance

23
The Range

• the difference between the largest and smallest
  values of a distribution

24
Find the range

• 10, 13, 17, 17, 18
• The range largest minus smallest
• 18 minus 10 8

25
The Standard Deviation

• a measure of the average variation of the data
  entries from the mean

26
Standard deviation of a sample
mean of the sample
n sample size
27
To calculate standard deviation of a sample

• Calculate the mean of the sample.
• Find the difference between each entry (x) and
  the mean. These differences will add up to zero.
• Square the deviations from the mean.
• Sum the squares of the deviations from the
  mean.
• Divide the sum by (n - 1) to get the variance.
• Take the square root of the variance to get the
  standard deviation.

28
The Variance

• the square of the standard deviation

29
Variance of a Sample
30
Find the standard deviation and variance
x 30 26 22
4 0 -4
16 0 16 ___
Sum 0
78
32
mean 26
31
The variance
32 2 16
32
The standard deviation
s
33
Find the mean, the standard deviation and variance
Find the mean, the standard deviation and variance
x 4 5 5 7 4
-1 0 0 2 -1
1 0 0 4 1
mean 5
25
6
34
The mean, the standard deviation and variance
Mean 5
35
Sum of Squares

• SSx

36
Computation formula for sample standard deviation
37
To find S x2

• Square the x values, then add.

38
To find ( S x ) 2
Sum the x values, then square.
39
Use the computing formulas to find s and s2
x 4 5 5 7 4
x2 16 25 25 49 16
n 5 (Sx) 2 25 2 625 Sx2 131 SSx
131 625/5 6 s2 6/(5 1) 1.5 s 1.22
131
25
40
Population Mean
41
Population Standard Deviation
42
Coefficient Of Variation

• A measurement of the relative variability (or
  consistency) of data.

43
CV is used to compare variability or consistency
A sample of newborn infants had a mean weight of
6.2 pounds with a standard deviation of 1 pound.
A sample of three-month-old children had a mean
weight of 10.5 pounds with a standard deviation
of 1.5 pound. Which (newborns or 3-month-olds)
are more variable in weight?
44
To compare variability, compare Coefficient of
Variation

• For newborns
• For 3-month-olds

• Higher CV more variable

CV 16 CV 14
Lower CV more consistent
45
Use Coefficient of Variation

• To compare two groups of data,
• to answer
• Which is more consistent?
• Which is more variable?

46
CHEBYSHEV'S THEOREM

• For any set of data and for any number k, greater
  than one, the proportion of the data that lies
  within k standard deviations of the mean is at
  least

47
CHEBYSHEV'S THEOREM for k 2
According to Chebyshevs Theorem, at least what
fraction of the data falls within k (k 2)
standard deviations of the mean?
At least
of the data falls within 2 standard deviations of
the mean.
48
CHEBYSHEV'S THEOREM for k 3
According to Chebyshevs Theorem, at least what
fraction of the data falls within k (k 3)
standard deviations of the mean?
At least
of the data falls within 3 standard deviations of
the mean.
49
CHEBYSHEV'S THEOREM for k 4
According to Chebyshevs Theorem, at least what
fraction of the data falls within k (k 4)
standard deviations of the mean?
At least
of the data falls within 4 standard deviations of
the mean.
50
Using Chebyshevs Theorem
A mathematics class completes an examination and
it is found that the class mean is 77 and the
standard deviation is 6. According to
Chebyshev's Theorem, between what two values
would at least 75 of the grades be?
51
Mean 77 Standard deviation 6
At least 75 of the grades would be in the
interval
77 2(6) to 77 2(6) 77 12 to 77 12 65 to
89
52
Percentiles

• For any whole number P (between 1 and 99), the
  Pth percentile of a distribution is a value such
  that P of the data fall at or below it.
• The percent falling above the Pth percentile will
  be (100 P).

53
Percentiles
40 of data
60 of data
54
Quartiles

• Percentiles that divide the data into fourths
• Q1 25th percentile
• Q2 the median
• Q3 75th percentile

55
Quartiles
Median Q2
Q1
Q3
Lowest value
Highest value
Inter-quartile range IQR Q3 Q1
56
Computing Quartiles

• Order the data from smallest to largest.
• Find the median, the second quartile.
• Find the median of the data falling below Q2.
  This is the first quartile.
• Find the median of the data falling above Q2.
  This is the third quartile.

57
Find the quartiles

• 12 15 16 16 17 18 22 22
• 23 24 25 30 32 33 33 34
• 41 45 51

The data has been ordered. The median is 24.
58
Find the quartiles
12 15 16 16 17 18 22 22 23 24 25 30
32 33 33 34 41 45 51
The data has been ordered. The median is 24.
59
Find the quartiles
12 15 16 16 17 18 22 22 23 24 25 30
32 33 33 34 41 45 51
For the data below the median, the median is
17. 17 is the first quartile.
60
Find the quartiles
12 15 16 16 17 18 22 22 23 24 25 30
32 33 33 34 41 45 51
For the data above the median, the median is
33. 33 is the third quartile.
61
Find the interquartile range

• 12 15 16 16 17 18 22 22
• 23 24 25 30 32 33 33 34
• 41 45 51

IQR Q3 Q1 33 17 16
62
Five-Number Summary of Data

• Lowest value
• First quartile
• Median
• Third quartile
• Highest value

63
Box-and-Whisker Plot

• a graphical presentation of the five-number
  summary of data

64
Making a Box-and-Whisker Plot

• Draw a vertical scale including the lowest and
  highest values.
• To the right of the scale, draw a box from Q1 to
  Q3.
• Draw a solid line through the box at the median.
• Draw lines (whiskers) from Q1 to the lowest and
  from Q3 to the highest values.

65
Construct a Box-and-Whisker Plot
12 15 16 16 17 18 22 22 23 24 25 30
32 33 33 34 41 45 51
Lowest 12 Q1 17 median 24 Q3 33 Highest
51
66
Box-and-Whisker Plot(Use a Horizontal not
Vertical Box Whisker)
Lowest 12 Q1 17 median 24 Q3 33 Highest
51