Download Elementary Statistics and Inference Elementary Statistics and

Elementary Statistics and
Inference
22S:025 or 7P:025
Lecture 4
1
Elementary Statistics and
Inference
22S:025 or 7P:025
Chapter 4
2
5.) Chapter Four
A.
Introduction
The histogram provides a general description of where
the scores are located, the “shape” of the density
distribution but not a good description of
distribution,
“spread/variation” of the scores, or the
location/concentration of the scores.
3
1
5.) Chapter Four (cont.)
„
The “center/location” of the scores is often described as
the average, or the median.
„
The standard deviation describes the “spread” around
the average
g score. A second index of the spread
p
of
scores in a histogram is the interquartile range.
„
The mean is the arithmetic average of the scores.
„
The median is the point on the score scale below which
50% of the scores fall.
4
5.) Chapter Four (cont.)
Examples:
28
24
20
16
%
12
8
4
0
5
6
7
8
9
10
11
Average = 8.00
Median = 8.00
Standard deviation = 1.66
5
5.) Chapter Four (cont.)
30
28
24
18
18
%
12
12
6
6
0
5
6
7
5
8 9 10 11 12 13 14 15
Average = 8.19
Median = 6.8
Standard deviation = 3.14
6
2
5.) Chapter Four (cont.)
The Average (Mean) is affected by every score, and is
pulled in the direction of the extreme scores.
When the distribution is symmetric, the mean and median
are the same. When the distribution is skewed – the mean
differs from the median.
If distribution is skewed right – median is less than mean.
If distribution is skewed left – median is larger than mean.
If distribution is symmetrical – median and mean are same.
7
5.) Chapter Four (cont.)
B.
The Average or Mean
„
Computation of Mean – find the sum of scores, then
divide by the number of scores.
Example: 9, 1, 2, 2, and 0
mean =
9 + 1 + 2 + 2 + 0 14
=
= 2.8
5
5
8
5.) Chapter Four (cont.)
„
On pages 58-60, the text provides data form the 1976-80
Health and Nutrition Examination Survey (Hanes) – a
representative cross section of 20,322 Americans aged
1-74.
Data were collected on:
‰ Demographics – age, education, income
‰ Physiological variables – height, weight, blood
pressure, etc.
‰ Dietary habits
‰ Prevalence of disease
‰ Levels of pesticides in blood.
9
3
5.) Chapter Four (cont.)
„
The plots of the average heights and weights by age in
years for the 2003-04 survey are shown in Figure 3
(page 59).
„
A symbol for the average or the mean is commonly
reported as x, and the sum of scores is represented by
ΣΧ .
„
So, x = ΣΧ / n = which means to find the sum of all the
scores, and then divide the sum by the total number of
scores.
10
5.) Chapter Four (cont.)
11
5.) Chapter Four (cont.)
Exercise Set A – pp. 60-61 assign 1, 3, 8
#4 N=10 avg=5 ft and 6 inches or 66 inches
This means the sum of their heights is 660 inches.
The 11th person is 6 feet 5 inches or 77 inches
inches.
The new sum would be 660 + 77 = 737 inches
737
= 67 inches or 5 feet
The new mean would be
11
and 7 inches.
12
4
5.) Chapter Four (cont.)
C.
Average and Histogram
„
See diagrams on pages 62-63 of text.
„
Figure 4 – page 62 shows histogram for 2,696 women
ages 18-74 from the HANES (Health and Nutrition
Examination Survey) from 2003-2004. The data for
weights are skewed to the right – i.e.,
i e the tail is to the
right. The average (mean) is pulled in the direction the
skew – because the extreme scores pull the average in
that direction.
For example,
Mean for 1, 3, 5, 7, 9 = 5
Mean for 1, 3, 5, 7, 90 = 21.2 (skewed right)
13
5.) Chapter Four (cont.)
14
5.) Chapter Four (cont.)
15
5
5.) Chapter Four (cont.)
„
If the data for a histogram are skewed to the left (tail is at
lower end of the histogram), the mean is pulled to the
left.
Mean for 10, 12, 14, 16, 18 = 14
Mean for 1, 5, 14, 16, 18 = 10.8
„
The Mean is the point of balance in a distribution of
scores – that is, the sum of the scores above the mean
is equal to the sum of scores below the mean – the
mean is a centroid.
Σ( x − x ) = 0
16
5.) Chapter Four (cont.)
The sum of the differences between each score and the
mean is always equal to zero.
Example:
ΣΧ = 25
1, 3, 5, 7, 9
ΣΧ
5
Σ( x − x ) = (1 − 5) + (3 − 5) + (5 − 5) + (7 − 5) + (9 − 5)
Σ( x − x ) = (−4) + (−2) + (0) + (2) + (4)
Σ( x − x ) = 0
mean = x = 5 =
The centroid (mean) for a seesaw (page 64) is such that it
can be balanced by two persons of different weights by
their distance from the centroid.
17
5.) Chapter Four (cont.)
18
6
5.) Chapter Four (cont.)
„
Median – as shown in histograms on page 64, the
median has 50% of the scores (area) below the median,
and 50% of the scores (area) above the median.
Symmetrical
y
histogram
g
– mean is same as median
Skewed Right histogram – median is less than mean
Skewed Left histogram – median is greater than mean
See Exercise Set B (page 65) – 1, 2, 3, 4
19
5.) Chapter Four (cont.)
„
Computing mean in a Histogram
x
f
f·x
19
1
19
18
2
36
17
3
51
16
4
64
15
5
75
14
6
84
13
4
52
12
3
36
11
2
22
N=30
Sum=439
x = 14.63
20
5.) Chapter Four (cont.)
Note: Again we assume each score has value of the
midpoint of score interval.
For example: the 3 scores of 17 would be evenly
distributed across the interval between
16.5 and 17.5
21
7
5.) Chapter Four (cont.)
„
Computing mean in a grouped distribution –
HISTOGRAM
x
x midpt
f
f · x midpt
18-20
19
2
38
15-17
16
3
48
12-14
13
4
52
9-11
10
6
60
6-8
7
3
21
3-5
4
2
8
N=20
Sum ≈227
x=
sum of scores
number of scores
227
= 11.35
20
x ≈ 11.35
x≈
Assume: Scores in an interval evenly distributed throughout
the score interval
f · midpt ≈ sum of scores in interval
22
5.) Chapter Four (cont.)
D.
Root-Mean-Square (RMS)
The RMS is an index in score scale units that
describes the spread or variation of the scores in a
histogram.
RMS = average of the squares of each score
= ΣX 2 / n
23
5.) Chapter Four (cont.)
Example:
3, 5, 7, 9, 11
average = x = mean = 7
9, 25, 49, 81, 121
RMS =
9 + 25 + 49 + 81 + 121
=
5
285
= 57 = 7.549
5
ƒ RMS is always slightly larger than the mean.
24
8