Transcript x - People Server at UNCW

Numerical descriptions of distributions
Describe the shape, center, and spread of a
distribution… for shape, see slide #6 below...
Center: mean and median
Spread: range, IQR, standard deviation
We treat these as aids to understanding the
distribution of the variable at hand…
We'll start with the mean: The mean is often called
the "average" and is in fact the arithmetic
average ("add all the values and divide by the
number of observations").
woman
(i)
height
(x)
woman
(i)
height
(x)
i=1
x1= 58.2
i = 14
x14= 64.0
i=2
x2= 59.5
i = 15
x15= 64.5
i=3
x3= 60.7
i = 16
x16= 64.1
i=4
x4= 60.9
i = 17
x17= 64.8
i=5
x5= 61.9
i = 18
x18= 65.2
i=6
x6= 61.9
i = 19
x19= 65.7
i=7
x7= 62.2
i = 20
x20= 66.2
i=8
x8= 62.2
i = 21
x21= 66.7
i=9
x9= 62.4
i = 22
x22= 67.1
i = 10
x10= 62.9
i = 23
x23= 67.8
i = 11
x11= 63.9
i = 24
x24= 68.9
i = 12
x12= 63.1
i = 25
x25= 69.6
i = 13
x13= 63.9
n=25
S=1598.3
Mathematical notation:
x1  x2  ...  xn
x
n
1 n
x   xi
n i 1
1598.3
x
 63.9
25
Learn right away how to get the mean with calculators & JMP
Your numerical summary must be meaningful!
Height of 25 women in a class
x  63.9
Here the shape of
the distribution is
wildly irregular.
Why?
Could we have
more than one
plant species or
phenotype?
The distribution of women’s
heights appears symmetrical. The
mean is a good numerical
summary.
x  69.6
Height of Plants by Color
x  63.9
5
x  70.5
x  78.3
red
Number of Plants
4
pink
blue
3
2
1
0
58
60
62
64
66
68
70
72
74
76
78
80
82
Height in centimeters
A single numerical summary here would not make sense.
84
• The Median (M) is often called the "middle" value and is the
value at the midpoint of the observations when they are
ranked from smallest to largest value….
– arrange the data from smallest to largest
– if n is odd then the median is the single observation in the center (at
the (n+1)/2 position in the ordering)
– if n is even then the median is the average of the two middle
observations (at the (n+1)/2 position; i.e., in between…)
In Table 1.10 (1.2,1/11),
calculate the mean and
median for the 2-seater
cars' city m.p.g. to see that
the mean is more sensitive
to outliers than the
median… use JMP-get data
from the eBook…
Skewness
Mode
=
Mean
=
Median
SYMMETRIC
Mean
Mode
Median
SKEWED LEFT
(negatively)
Mean
Mode
Median
SKEWED RIGHT
(positively)
Mean and median of a distribution with outliers
Percent of people dying
x  3.4
x  4.2
Without the outliers
With the outliers
The mean is pulled to the
The median, on the other hand,
right a lot by the outliers
is only slightly pulled to the right
(from 3.4 to 4.2).
by the outliers (from 3.4 to 3.6).
Impact of skewed data
Mean and median of a symmetric
Disease X:
x  3.4
M  3.4
Mean and median are the same.
… and a right-skewed distribution
Multiple myeloma:
x  3.4
M  2.5
The mean is pulled toward
the direction of the skew.
Spread: percentiles, quartiles (Q1 and Q3), IQR,
5-number summary (and boxplots), range,
standard deviation
pth percentile of a variable is a data value such that
p% of the values of the variable are less than or
equal to it.
the lower (Q1) and upper (Q3) quartiles are special
percentiles dividing the data into quarters
(fourths). get them by finding the medians of the
lower and upper halfs of the data
IQR = interquartile range = Q3 - Q1 = spread of
the middle 50% of the data. IQR is used with
the so-called 1.5*IQR criterion for outliers - know
this!
Measure of spread: the quartiles
The first quartile, Q1, is the value in the
sample that has 25% of the data less
than or equal to it ( it is the median of
the lower half of the sorted data,
excluding M).
M = median = 3.4
The third quartile, Q3, is the value in the
sample that has 75% of the data less
than or equal to it ( it is the median of
the upper half of the sorted data,
excluding M).
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
1
2
3
4
5
6
7
1
2
3
4
5
1
2
3
4
5
6
7
1
2
3
4
5
0.6
1.2
1.5
1.6
1.9
2.1
2.3
2.3
2.5
2.8
2.9
3.3
3.4
3.6
3.7
3.8
3.9
4.1
4.2
4.5
4.7
4.9
5.3
5.6
6.1
Q1= first quartile = 2.2
Q3= third quartile = 4.35
Five-number summary and boxplot
6
5
4
3
2
1
6
5
4
3
2
1
6
5
4
3
2
1
6
5
4
3
2
1
6.1
5.6
5.3
4.9
4.7
4.5
4.2
4.1
3.9
3.8
3.7
3.6
3.4
3.3
2.9
2.8
2.5
2.3
2.3
2.1
1.9
1.6
1.5
1.2
0.6
Largest = max = 6.1
BOXPLOT
7
Q3= third quartile
= 4.35
M = median = 3.4
6
Years until death
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
5
4
3
2
1
Q1= first quartile
= 2.2
Smallest = min = 0.6
0
Disease X
Five-number summary:
min Q1 M Q3 max
Boxplots for skewed data
Years until death
Comparing box plots for a normal
and a right-skewed distribution
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Boxplots remain
true to the data and
depict clearly
symmetry or skew.
Disease X
Multiple Myeloma
5-number summary: min. , Q1, median, Q3, max
when plotted, the 5-number summary is a boxplot we can also
do a modified boxplot to show outliers (mild and extreme).
Boxplots have less detail than histograms and are often
used for comparing distributions… e.g., Fig. 1.19, p.37 and
below...
Figure 1.19
Introduction to the Practice of Statistics, Sixth Edition
© 2009 W.H. Freeman and Company
6
5
4
3
2
1
6
5
4
3
2
1
6
5
4
3
2
1
6
5
4
3
2
1
7.9
6.1
5.3
4.9
4.7
4.5
4.2
4.1
3.9
3.8
3.7
3.6
3.4
3.3
2.9
2.8
2.5
2.3
2.3
2.1
1.9
1.6
1.5
1.2
0.6
8
7
Q3 = 4.35
6
Years until death
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
Distance to Q3
7.9 − 4.35 = 3.55
5
Interquartile range
Q3 – Q1
4.35 − 2.2 = 2.15
4
3
2
1
Q1 = 2.2
0
Disease X
Individual #25 has a value of 7.9 years, which is 3.55 years
above the third quartile. This is more than 3.225 years, 1.5 *
IQR. Thus, individual #25 is an outlier by our 1.5 * IQR rule.
Definition, pg 40–41
Introduction to the Practice of Statistics, Sixth Edition
© 2009 W.H. Freeman and Company
Look at Example 1.19 on page 41 (1.2, 8/11) – see
Fig. 1.21 for a graph of deviations from the mean...
metabolic rates for 7 men in a dieting study: 1792,
1666, 1362, 1614, 1460, 1867, 1439. Mean=1600
cals., s=189.24 calories.
Figure 1.21
Introduction to the Practice of Statistics, Sixth Edition
© 2009 W.H. Freeman and Company
Be sure you know how to compute the standard
deviation with JMP since it’s almost never done
by hand with the previous page’s formula...
Put the metabolic rates into a JMP table and
analyze…
why do we square the deviations? - two technical
reasons that we'll see when we discuss the normal
distribution in the next section…
why do we use the standard deviation (s) instead of
the variance (s2)? s2 has units which are the
squares of the original units of the data…
why do we divide by n-1 instead of n? n-1 is called
the number of degrees of freedom; since the sum
of the deviations is zero, the last deviation can
always be found if we know n-1 of them …
which measure of spread is best? 5-number
summary is better than the mean and s.d. for
skewed data - use mean & s.d. for symmetric data
What should you use, when, and why?
Arithmetic mean or median?
• Middletown is considering imposing an income tax on citizens. City
hall wants a numerical summary of its citizens income to estimate
the total tax base.
– Mean: Although income is likely to be right-skewed, the city
government wants to know about the total tax base.
• In a study of standard of living of typical families in Middletown, a
sociologist makes a numerical summary of family income in that city.
– Median: The sociologist is interested in a “typical” family and
wants to lessen the impact of extreme incomes.
• Finish reading section 1.2
• Be sure to go over the Summary at the end
of each section and know all the
terminology
• Do # 1.56, 1.62-1.64, 1.67, 1.69, 1.75-1.77
(Mean/Median Applet), 1.78, 1.79
• use JMP for any problem requiring more
than very simple computations…