Title: Understanding Basic Statistics Third Edition By Brase and Brase Prepared by: Lynn Smith Gloucester C
1Understanding Basic StatisticsThird EditionBy
Brase and BrasePrepared by Lynn
SmithGloucester County College
- Chapter Three
- Averages and Variation
2Measures of Central Tendency
3The Mode
- the value that occurs most frequently in a data
set
4Find the mode
- 6, 7, 2, 3, 4, 6, 2, 6
- The mode is 6.
5Find the mode
- 6, 7, 2, 3, 4, 5, 9, 8
- There is no mode for this data.
6The Median
- the central value of an ordered distribution
7To 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
8Find the median
- Data 5, 2, 7, 1, 4, 3, 2
- Rearrange 1, 2, 2, 3, 4, 5, 7
The median is 3.
9Find 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
10The Mean
- The mean of a collection of data is found by
- summing all the entries
- dividing by the number of entries
11Find the mean
6, 7, 2, 3, 4, 5, 2, 8
12Sigma Notation
- The symbol S means sum the following.
- S is the Greek letter (capital) sigma.
13Notations for mean
- Population mean
- Greek letter (mu)
14Number 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.
15Sample mean
16Population mean
17Resistant Measure
- a measure that is not influenced by extremely
high or low data values
18Which is less resistant?
- The mean is less resistant. It can be made
arbitrarily large by increasing the size of one
value.
19Trimmed Mean
- a measure of center that is more resistant than
the mean but is still sensitive to specific data
values
20To 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.
21Compute 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
22Measures of Variation
- Range
- Standard Deviation
- Variance
23The Range
- the difference between the largest and smallest
values of a distribution
24Find the range
- 10, 13, 17, 17, 18
- The range largest minus smallest
- 18 minus 10 8
25The Standard Deviation
- a measure of the average variation of the data
entries from the mean
26Standard deviation of a sample
mean of the sample
n sample size
27To 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.
28The Variance
- the square of the standard deviation
29Variance of a Sample
30Find the standard deviation and variance
x 30 26 22
4 0 -4
16 0 16 ___
Sum 0
78
32
mean 26
31The variance
32 2 16
32The standard deviation
s
33Find 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
34The mean, the standard deviation and variance
Mean 5
35Sum of Squares
36Computation formula for sample standard deviation
37To find S x2
- Square the x values, then add.
38To find ( S x ) 2
Sum the x values, then square.
39Use 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
40Population Mean
41Population Standard Deviation
42Coefficient Of Variation
- A measurement of the relative variability (or
consistency) of data.
43CV 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?
44To compare variability, compare Coefficient of
Variation
- For newborns
- For 3-month-olds
CV 16 CV 14
Lower CV more consistent
45Use Coefficient of Variation
- To compare two groups of data,
- to answer
- Which is more consistent?
- Which is more variable?
46CHEBYSHEV'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
47CHEBYSHEV'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.
48CHEBYSHEV'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.
49CHEBYSHEV'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.
50Using 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?
51Mean 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
52Percentiles
- 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).
53Percentiles
40 of data
60 of data
54Quartiles
- Percentiles that divide the data into fourths
- Q1 25th percentile
- Q2 the median
- Q3 75th percentile
55Quartiles
Median Q2
Q1
Q3
Lowest value
Highest value
Inter-quartile range IQR Q3 Q1
56Computing 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.
57Find 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.
58Find 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.
59Find 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.
60Find 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.
61Find 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
62Five-Number Summary of Data
- Lowest value
- First quartile
- Median
- Third quartile
- Highest value
63Box-and-Whisker Plot
- a graphical presentation of the five-number
summary of data
64Making 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.
65Construct 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
66Box-and-Whisker Plot(Use a Horizontal not
Vertical Box Whisker)
Lowest 12 Q1 17 median 24 Q3 33 Highest
51