1 TEST REVIEW - Continued
BPS - 5th Ed.
Chapter 2
1
Three characteristics of a quantitative variable's
distribution:
Shape Visually via graphs. Chapter 1
Center (Typical Value) Numeric summary.
Chapter 2
Spread (Dispersion) Numeric summary.
Appropriate measures of center and spread
depends upon the distribution's shape.
BPS - 5th Ed.
Chapter 2
2
Review: Populations vs.
S versus analyzing samples.
Analyzing populations amples
For populations:
We know all of the data.
Descriptive measures of populations are called
parameters.
Parameters are often written using Greek letters
( ).
For samples:
We know only part of the entire data.
Descriptive measures of samples are called x
sEd.
BPS - 5th tatistics.
Chapter 2
3
Discriptive Measures: Numerical
Summaries
Center of the data: Numeric values that
represent the average or typical value of a
quantitative variable. Examples of Measures
of Central Tendancy
mean
median
BPS - 5th Ed.
Chapter 2
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Discriptive Measures: Numerical
Summaries (cont.)
Variation or Measures of Dispersion:
Numeric values that represent the degree to
which the values are spread out.
range
quartiles (interquartile range)
variance
standard deviation
BPS - 5th Ed.
Chapter 2
5
Central Tendency: Mean
The mean (arithmetic mean) of a variable is often what
people mean by the average add up all the values
and divide by the number of measurements in the data
set.
To compute the arithmetic mean of:
6, 1, 5
Add up the three numbers and divide by 3.
(6 + 1 + 5) / 3 = 4.0
The arithmetic mean is 4.0, one more decimal place than the
General Rounding Rule for
data.
Reporting Statistics
One interpretation: The arithmetic mean can be thought of as the
center of gravity where the yardstick balances.
BPS - 5th Ed.
Chapter 2
6
Central Tendency: Mean
The mean, more accurately
known as the arithmetic
mean, is an arithmetic
average of the elements of
the data set.
n
The mean of a sample
of n measurements is
x
denoted by
and
equals:
x=
xi
i =1
statistic
n = Sample Size
n
N
=
If the data are from a
population, the mean is 2
BPS - 5th Ed.
Chapter
xi
i =1
N
parameter
N = Population Size
7
Example A: Finding the mean
Sample data set: 1, 4, 5, 10
x1 = 1, x2 = 4, x3 = 5, x4 = 10
n
x=
x=
xi
i =1
n
1 + 4 + 5 + 10
4
x = 5.0
BPS - 5th Ed.
Chapter 2
8
Central Tendency: Median (M)
A resistant measure of the datas center
With ordered data, that is data arranged in
ascending order, at least half of the ordered
values are less than or equal to the median
value
With ordered data, that is data arranged in
ascending order, at least half of the ordered
values are greater than or equal to the median
value
If n is odd, the median is the middle ordered data value
If n is even, the median is the average of the two
BPS - 5th Ed.
Chapter
9
middle ordered data values 2
Median (M): Finding the position of the
median in your ordered data set
Location of the median: L(M) = (n+1)/2 ,
where n = sample size.
Example: Take this ordered data set:
x1, x2, x3, .x25
x1< x2<< x25
The position of the Median would be the
(25+1)/2 = 13th ordered value. That is the
Median = x13
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Chapter 2
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Median
Example 1 data: 2 4 6
[ x1 = 2, x2 = 4, x3 = 6 ]
L(M) = (3+1)/2 =2
Median: M = 4.0
Example 2 data: 2 4 6 8
[ x1 = 2, x2 = 4, x3 = 6, x4 = 8 ]
L(M)= (4+1)/2 = 2.5
so we must average data values x2 and x3 to get the
median
M =(4 +6)/2 = 5.0
Example 3 data: 6 2 4
BPS - 5th Ed.
Why is the Median not equal to 2.0?
Chapter 2
11
Median
Example 3 data: 6 2 4
The Median is not equal to 2.0 because we did not
order our data. After ordering our data, it is just like
example 1.
Example 4 data: 6 ,1, 11, 2, 11
Arrange in order: 1, 2, 6, 11, 11
L(M) = 3 so x3 is the median. M = 6.0
Example 5 data: 68, 71, 74, 77, 82, 84, 88, 90
Already in order
L(M) = 4.5, so the median is the average of x4 and x5
Median: M = (77+82)/2 = 79.5
M = 79.5
BPS - 5th Ed.
Chapter 2
12
Comparing the Mean & Median
The mean and median of data from a
symmetric distribution should be close
together. The actual (true) mean and median
of a symmetric distribution are exactly the
same.
In a skewed distribution, the mean is farther
out in the long tail than is the median [the
mean is pulled in the direction of the
possible outlier(s)].
The next slide will illustrate how knowing the
mean and median can give clues about the
shape of a distribution.
BPS - 5th Ed.
Chapter 2
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Comparing the Mean & Median
Since the mean and the median are often different.
This difference gives us clues about the shape of
the distribution. (symmetric?, skewed left?,
skewed right?)
Symmetric mean will usually be close to the median.
Skewed left mean will usually be smaller than the median.
Skewed right mean will usually be larger than the median
BPS - 5th Ed.
Chapter 2
14
Resistant Statistic
What if one value is extremely different from the
others?
Example: What if we made a mistake and
6, 1, 2
was recorded as
6000, 1, 2 ?
The mean is now ( 6000 + 1 + 2 ) / 3 = 2001.0
The median is still 2.0
Conclusion: The median is resistant to extreme values
while the mean is not resistant.
When data has extreme values (i.e., unusually large or small
values) or is skewed, median preferred to mean because it is
more representative of the "typical" value of the variable.
BPS - 5th Ed.
Chapter 2
15
Summary: Central Tendency
Mean
Center of gravity.
Useful for roughly symmetric quantitative data with no
extreme values.
Median
Splits the data into halves.
Useful for highly skewed quantitative data (or data
with extreme values).
BPS - 5th Ed.
Chapter 2
16
Spread, or Variability
If all values are the same, then they all
equal the mean. There is no variability.
Variability exists when some values are
different from (above or below) the mean.
BPS - 5th Ed.
Chapter 2
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Measuring Spread or Variability
Measures of spread, or measures of dispersion:
Numerical values that represent the degree to
which the values are spread out.
The range
The quartiles
First quartile or 25th percentile
Second quartile or 50th percentile or Median
Third quartile or 75th percentile
The variance
The standard deviation
BPS - 5th Ed.
Chapter 2
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Range
The range of a variable is the
largest data value (called the maximum) minus
the smallest data value (called the minimum).
Example 6: Compute the range of 6, 1, 2, 6, 11, 7,
3, 3
The largest value is 11.
The smallest value is 1.
Subtracting the two 11 1 = 10 the
range is 10.0, one more decimal place than
the data.
General Rounding Rule for Reporting Statistics
BPS - 5th Ed.
Chapter 2
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Quartiles
Three numbers which divide the ordered
data into four equal sized groups.
Q1 has 25% of the data below it.
Q2 has 50% of the data below it. (Median)
Q3 has 75% of the data below it.
BPS - 5th Ed.
Chapter 2
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Quartiles
Uniform Distribution
1st Qtr
BPS - 5th Ed.
Q1 2nd Qtr Q2
3rd Qtr
Chapter 2
Q3
4th Qtr
21
Obtaining the Quartiles
Order the data.
For Q2, just find the median.
For Q1, look at the lower half of the data
values, those to the left of the median
location; find the median of this lower half.
For Q3, look at the upper half of the data
values, those to the right of the median
location; find the median of this upper
half.
BPS - 5th Ed.
Chapter 2
22
Example 7
Even Data Set: 5 25 7 23 10 11 15 21 18 20
1. Put Data Set in order and find location of
median:
5 7 10 11 15 18 20 21 23 25
L(M) = (10+1)/2 = 5.5 so the median or 2nd
quartile is the average of the 5th and 6th data
value
2. M = 16.5
3. There are 5 data points below and above
median.
4. the Find median of the first 5 data points and
t5th Ed. is the 1st quartile.Chapter 2
hat
BPS 23
Example 7 (cont.)
Data Set: 5 7 10 11 15 18 20 21 23 25
To find the 1st Quartile, we must find the
median of the first half of the data (data to
the left of the Median). The L(M) of the
first 5 data points is
L(M) = (5+1)/2 = 3
so the 1st quartile is x3 = 10.0
BPS - 5th Ed.
Chapter 2
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Example 7 (cont.)
Data Set: 5 7 10 11 15 18 20 21 23 25
To find the 3rd Quartile, we must find the
median of the second half of the data
(data to the right of the Median). The L(M)
of the last 5 data points is
L(M) = (5+1)/2 = 3
so the 3rd quartile is 3rd data point in that
set, so 3rd quartile is 21.0
BPS - 5th Ed.
Chapter 2
25
Interquartile Range (IQR)
The interquartile range (IQR) is the difference
between the third and first quartiles.
IQR = Q3 Q1
The IQR is a resistant measure of dispersion.
IQR
Interquartile
Range
BPS - 5th Ed.
Chapter 2
26
Five-Number Summary
Five-Number Summary gives a
concise description of the
distribution of a variable:
Smallest value (Min)
First quartile (Q1 or P25)
Median (M or Q2 or P50)
Third quartile (Q3 or P75)
Largest value (Max)
The first quartile and the
third quartile:
Information about the
spread of the data (IQR).
Resistant.
BPS - 5th Ed.
The smallest value (min) and
the largest value (max):
Information about the tails
of the data.
Not resistant.
The median:
Information about the
center of the data.
Resistant.
Chapter 2
27
Boxplot
A boxplot is a graphical representation of the
five-number summary
Central box spans Q1 and Q3.
A line in the box marks the median M (Q2).
Lines extend from the box out to the minimum
and maximum. (These lines are sometimes
called whiskers)
BPS - 5th Ed.
Chapter 2
28
Boxplot of Example 9: Weight Data
Q1
100
275
BPS - 5th Ed.
125
M
150
Q3
175
max
200
225
250
Weight
Chapter 2
29
Boxplot
WARNING: When Using MINITAB, the
whiskers show the minimum and maximum
values within what is called the lower and upper
fences (Numbers greater than the upper fence
and smaller than the lower fence are things
called outliers) not the min and max of the data
set necessarily. Asterisks are used to indicate
any outliers.
We will talk about what an outlier is and a way
we can check our data for outliers.
BPS - 5th Ed.
Chapter 2
30
Outliers
Outliers are extreme observations in the data. They
are values that are significantly too high or too
low, based on the spread of the data.
Outliers should be identified and investigated.
Outliers could be:
Chance occurrences
Measurement errors
Data entry errors
Sampling errors
Outliers are not necessarily invalid data.
One way to check for outliers uses the quartiles.
BPS - 5th Ed.
Chapter 2
31
Outliers (cont.)
Fence Rule for checking for outliers using the
quartiles:
Calculate lower and upper fences:
Lower fence = LF = Q1 (1.5 IQR)
Upper fence = UF = Q3 + (1.5 IQR)
Values (strictly) less than the lower fence or
(strictly) greater than the upper fence could be
considered outliers.
BPS - 5th Ed.
Chapter 2
32
Boxplot in MINITAB of Example 7
The MINITAB commands that
went with producing this boxplot
(Data was in column 1):
MTB> boxplot c1;
SUBC> transpose.
From the boxplot produced from MINITAB, we see that
there are no outliers. (No asterisks)
BPS - 5th Ed.
Chapter 2
33
Boxplot in MINITAB of Altered Weight Data
The MINITAB commands that
went with producing this boxplot
(Data was in column 2):
MTB> boxplot c2;
SUBC> transpose.
From the boxplot produced from MINITAB, we see that
there are outliers. The left whisker goes to the minimum
data value, but the right whisker goes to the upper fence.
And the asterisks goes to the maximum data value (or any
of the outliers, in this example there is only one outlier and
it happens to be the maximum value in the data set).
BPS - 5th Ed.
Chapter 2
34
Example from Text: Boxplots
BPS - 5th Ed.
Chapter 2
35
Variance and Standard
Deviation
Recall that variability exists when some
values are different from (above or
below) the mean.
Each data value has an associated
deviation from the mean.
BPS - 5th Ed.
Chapter 2
36
Variance
The variance is based on the deviation from the
mean. (How far is each observation from the
( i )
typicalxvalue?) for populations
( xi x )
for samples
To treat positive differences and negative these
The larger
2 for populations
(x )
values,
differences, we square the deviations: the further
i
( xi
BPS - 5th Ed.
x ) 2 for samples
Chapter 2
observations are
from the mean,
i.e., the more
spread out the
37
data values.
Population Variance
The population variance (denoted 2) of a variable is
the sum of these squared deviations divided by the
number in the population (i.e., population size ).
(In other words we are calculating the average of the
squared distance of each of the data points.)
2 =
( xi ) 2
N
( x1 )2 + ( x2 )2 + L + ( x N )2
=
N
Note: If you are calculating by hand, use as many decimal places as
allowed by your calculator.
BPS - 5th Ed.
Chapter 2
38
Sample Variance
The sample variance (denoted s2) of a variable is the sum of these
squared deviations divided by one less than the number in the
sample (i.e., sample size minus 1).
s2 =
( xi x )2
n1
( x1 x )2 + ( x2 x )2 + L + ( xn x )2
=
n1
We say that this statistic has n 1 degrees of freedom.
Note: For accuracy when calculating by hand, use as many
decimal places as allowed by your calculator.
Why do we use different formulas for the population variance and the
sample variance? See page 51 in your book.
BPS - 5th Ed.
Chapter 2
39
Standard Deviation
The standard deviation is the square root of the
variance.
The population standard deviation
Is the square root of the population variance (2).
Is represented by .
The sample standard deviation
Is the square root of the sample variance (s2).
Is represented by s.
To avoid round-off error, never use the rounded value of the
variance to compute the standard deviation.
The variance and standard deviation are not
resistant measures of dispersion.
BPS - 5th Ed.
Chapter 2
40
Deviations
what is a typical deviation from the mean?
(standard deviation)
small values of this typical deviation indicate
small variability in the data
large values of this typical deviation indicate
large variability in the data
BPS - 5th Ed.
Chapter 2
41
Variance and Standard Deviation
Example from Text
Metabolic rates of 7 men (cal./24hr.) :
1792 1666 1362 1614 1460 1867 1439
x
1792 + 1666 + 1362 + 1614 + 1460 + 1867 + 1439
=
7
11,200
=
7
= 1600
BPS - 5th Ed.
Chapter 2
42
Variance and Standard Deviation
Example from Text
Observations
xi
Deviations
Squared deviations
( xi x )
xi x
2
1792
17921600 = 192
1666
1666 1600 =
1362
1362 1600 = -238
1614
1614 1600 =
1460
1460 1600 = -140
(-140)2 = 19,600
1867
1867 1600 = 267
(267)2 = 71,289
1439
1439 1600 = -161
(-161)2 = 25,921
sum =
BPS - 5th Ed.
66
14
0
Chapter 2
(192)2 = 36,864
(66)2 =
4,356
(-238)2 = 56,644
(14)2 =
196
sum = 214,870
43
Variance and Standard Deviation
Example from Text
214,870
s=
= 35,811.67
7 1
2
s = 35,811.67 = 189.24 calories
BPS - 5th Ed.
Chapter 2
44
Choosing Measures of Center and Spread
(A Summary)
Outliers affect the values of the mean and
standard deviation.
The five-number summary should be used to
describe center and spread for skewed
distributions, or when outliers are present.
Use the mean and standard deviation for
reasonably symmetric distributions that are
free of outliers.
BPS - 5th Ed.
Chapter 2
45
Statistical Problem
BPS - 5th Ed.
Chapter 2
46