Descriptive Statistics
Healey Chapters 3 and 4
nd
(2 Cdn Ch. 3)
Measures of Central Tendency
And Dispersion
Measures of Central Tendency
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1. Mode = can be used for any kind of data
but only measure of central tendency for
nominal or qualitative data.
Formula: value that occurs most often or the
category or interval with highest frequency.
Note: Omit Formula 3.1 Variation Ratio in
Healey and Prus 2nd Cdn.
Example for Nominal Variables:
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Religion
Catholic
Protestant
Jewish
Muslim
Other
None
frequency
17
4
2
1
9
8
cf
17
21
23
24
33
41
proportion
.41
.10
.05
.02
.22
.20
%
41
10
5
2
9
20
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Total41
1.00
100%
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Central Tendency: MODE = largest category = Catholic
Cum%
41
51
56
58
80
100
Central Tendency (cont.)
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2. Median = exact centre or middle of
ordered data. The 50th percentile.
Formula:
Array data.
When sample even #, median falls halfway
between two middle numbers.
To calculate: find(n/2)and (n/2)+1, and divide
the total by 2 to find the exact median.
When sample is odd #, median is exact
middle (n+1) /2)
Example for Raw Data:
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Suppose you have the following set of test
scores:
66, 89, 41, 98, 76, 77, 69, 60, 60, 66, 69, 66,
98, 52, 74, 66, 89, 95, 66, 69
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1. Array data:
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98 98 95 89
69 66 66 66
89
77
76
74
66
66
60
60
N = 20 (N is even)
69
52
69
41
To calculate:
- find middle numbers(n/2)+(n/2 )+1
- add together the two middle numbers
- divide the total by 2
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First middle number: (20/2) = the 10th number
2nd middle number: (20/2)+1 = the 11th
Look at data:
the middle numbers are 69 and 69
The median would be (69+69)/2 = 69
Median for Aggregate (grouped) Data
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This formula is shown in Healey 1st Cdn
Edition and in Healey 8e but NOT in 2nd Cdn
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We will NOT COVER this one!
Properties of median:
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- for numerical data at interval or ordinal level
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-"balance point“
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-not affected by outliers
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-median is appropriate when distribution is
highly skewed.
3. Mean for Raw Data
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The mean is the sum of measurements /
number of subjects
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Formula: (X-bar)
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Data (from above):
66, 89, 41, 98, 76, 77, 69, 60, 60, 66, 69, 66,
98, 52, 74, 66, 89, 95, 66, 69
= ΣXi / N
Example for Mean
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Formula:
= ΣXi / N
= 1446 / 20
= 72.3
The mean for these test scores is 72.3
Mean for Aggregate (Grouped) Data
(Note: 1st Cdn. Edition: use this formula!
Omitted in 2nd Cdn. Ed. but covered in class)
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To calculate the mean for grouped data, you
need a frequency table that includes a
column for the midpoints, for the product of
the frequencies times the midpoints (fm).
Formula:
= Σ (fm)
N
Frequency table:
Score
f
m*
41-50
1
45.5
51-60
3
55.5
61-70
8
65.5
71-80
3
75.5
81-90
2
85.5
91-100
3
95.5
N = 20
Σ (fm) =
* Find midpoints first
(fm)
45.5
166.5
524
226.5
171
286.5
1420
Calculating Mean for Grouped Data:
Formula:
= Σ (fm)
N
= 1420 / 20
= 71
The mean for the grouped data is 71.
Properties of the Mean:
- only for numerical data at interval level
- "balance point“
- can be affected by outliers = skewed distribution
- tail becomes elongated and the mean is pulled in
direction of outlier.
Example…
no outlier:
$30000, 30000, 35000, 25000, 30000 then mean = $30000
but if outlier is present, then:
$130000, 30000, 35000, 25000, 30000 then mean = $50000
(the mean is pulled up or down in the direction of the outlier)
NOTE:
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When distribution is symmetric,
mean = median = mode
For skewed, mean will lie in direction of skew.
i.e. skewed to right,
mean > median (positive skew)
skewed to left,
median > mean (negative skew)
Measures of Dispersion
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Describe how variable the data are.
i.e. how spread out around the mean
Also called measures of variation or
variability
Variability for Non-numerical Data
(Nominal or Ordinal Level Data)
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Measures of variability for non-numerical
nominal or ordinal) data are rarely used
We will not be covering these in class
Omit Formula 4.1 IQV in Healey and Prus
1st Canadian Edition and in Healey 8e
Omit Formula 3.1 Variation Ratio in Healey
and Prus 2nd Canadian Edition
2. Range (for numerical data)
Range = difference between largest and
smallest observations
i.e. if data are $130000, 35000, 30000, 30000,
30000, 30000, 25000, 25000
then range = 130000 - 25000 = $105000
Interquartile Range (Q):
-
-
This is the difference between the 75th and the 25th
percentiles (the middle 50%)
Gives better idea than range of what the middle of
the distribution looks like.
Formula:
Q = Q3 - Q1 (where Q3 = N x .75,
and Q1 = N x .25)
Using above data: Q
= Q3 - Q1 = (6th – 2nd case)
= $30000-25000 =$5000
The interquartile range (Q) is $5000.
3. Variance and Standard Deviation:
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For raw data at the interval/ratio level.
Most common measure of variation.
The numerator in the formula is known as
the sum of squares, and the denominator is
either the population size N or the sample
size n-1
The variance is denoted by S2 and the
standard deviation, which is the square root
of the variance, by S
Definitional Formula for Variance and
Standard Deviation:
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Variance:
s2 = Σ (xi -
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S.D.:
s =
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A working formula (the one you use) for s.d
is:
1 N ∑ Xi2 - ( ∑ Xi ) 2
N
)2 / N
Example for S and
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1.
2.
3.
4.
2
S
:
Data: 66, 89, 41, 98, 76, 77, 69, 60, 60, 66,
69, 66, 98, 52, 74, 66, 89, 95, 66, 69
Find ∑ Xi2 : Square each Xi and find total.
Find (∑ Xi)2 : Find total of all Xi and square.
Substitute above and N into formula for S.
For S2 , simply square S.
S = 14.76
S2 = 217.91
Another working formula for the standard
deviation:
S
X
N
2
i
X
2
Note that the definitional formula for s.d. is
not practical for use with data when N>10.
The working formulae should be used instead.
All three formulae give exactly the same result.
Properties of S:
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always greater than or equal to 0
the greater the variation about mean,
the greater S is
n-1 (corrects for bias when using sample
data.) S tends to underestimate the
population s.d. so to correct for this, we use
n-1. The larger the sample size, the smaller
difference this correction makes. When
calculating the s.d. for the whole population,
use N in the denominator.
NOTE:
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σ, N and Mu (µ) denote population
parameters
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s, n, x-bar (
) denote sample statistics
Remember the Rounding Rules!
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Always use as many decimal places as your
calculator can handle.
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Round your final answer to 2 decimal places,
rounding to nearest number.
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Engineers Rule: When last digit is exactly 5
(followed by 0’s), round the digit before the
last digit to nearest EVEN number.
Homework Questions
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Healey and Prus 1st Cdn. And Healey 8e:
#3.1, #3.5, #3.11 and 4.9, #4.15
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Healey and Prus 2nd Cdn.
#3.1, #3.5, #3.11 (compute s for 8 nations also), #3.15
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SPSS:
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Read the SPSS sections for Ch. 3 and 4 in 1st Cdn. Edition
and for Ch. 4 in 2nd Cdn. Edition
Try some of the SPSS exercises for practice