Transcript 36c6d9a31e04bad

Measures of Central Tendency
and
Dispersion
Preferred measures of central location & dispersion
Type of Distribution
Central location
Dispersion
Normal
Mean
SD
Skewed
Median
Inter-quartile range?
Exponential or
logarithmic
Mean
Median
?
NORMAL DISTRIBUTION
Frequency Distribution: the Normal Distribution
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
Bell-shaped: specific shape that can be defined as an
equation
Symmetrical around the mid point, where the greatest
frequency if scores occur
Asymptotes of the perfect curve never quite meet the
horizontal axis
Normal distribution is an assumption of parametric
testing
Frequency Distribution: Different Distribution shapes
Measure of Central Tendency
 Mean
 Median
 Mode
Mean

It is computed by summing up all the observations in the
variable & dividing the sum by the number of
observations.

Mean (Average) =

The mean is the most commonly used measure since it
takes into account each observation
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It is problems:
It considers all observation and it is affected by all
observations not preferred in the presence of dispersed
values like salaries.
Sum of the Observation values
Number of observations
Mean (Average)
 Mean (Average) =
 Sum of the Observation values
Number of observations
 In this observation set
(5, 3, 9, 7, 1, 3, 6, 8, 2, 6, 6)
 Sum = 56
Number of observations = 11
Mean = 5.1
Weighted Mean
Village
1
2
3
4
5
No. of Children
Mean age (month)
54
52
49
48
48
251
58.6
59.5
61.2
62.5
64.5
61.2
(n1 X x1) + (n2 X x2) + .....
Weighted Mean = --------------------------------------
N
Geometric Mean

Mean of a set of data measured on a logarithmic scale.

Logarithmic scale is used when data are not normally
distributed & follow an exponential pattern (1,2,4,8,16) or
logarithmic pattern (1/2,1/4,1/8…)
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Geometric mean equals:
Anti Log for average of sum log of the values
Or: Anti Log (1/n ∑ Log Xi)
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So to calculate the Geometric mean
1-calculate sum of the logarithm of each value
2-calculate average by dividing sum of Log values by
number of these values
3-calculating of the anti log will give the geometric mean

Geometric Mean
Sample
Dilution
Titre
1
1:4
4
2
1:256
256
3
1:2
2
4
1:16
16
5
1:64
64
6
1:32
32
7
1:512
512
Geometric Mean

Calculate the geometric mean:
1-Sum of Log (4, 256, 2, 16, 64, 32, 512) = 10.536
2. Average = 10.536 / 7 = 1.505
3- Anti Log average =32
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Accordingly geometric mean =32

Geometric mean is important in statistical analysis of data
following the previous described distribution such as sero
survey where titer is calculated for different samples.
Median

Median: Value that divides a distribution into two equal
parts.

Arrange the observation by order
1,2,3,3,5,6,6,6,7,8,9.

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When the number is odd
Median = No. + 1 = 11+1 = 6
2
2
So, median is the 6th observations = 6
The median is the best measure when the data is skewed
or there are some extreme values
Median
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When number is even
1,2,3,3,5,6,6,6,7,8.
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Number of observations = 10
Median=
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5th observation + 6th observation
2
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5+6 = 11 = 5.5
2
2
Mode
 Mode: The most frequent value.
(5, 3, 9, 7, 1, 6, 8, 2, 6, 6)
 " 6" is the most frequent value. Bimodal
distribution is referred to presence of two most
frequent values.
 If all values are different there is no mode.
 Not useful when there are several values that
occur equally often in a set
Central Tendencies & Distribution Shape
 The mean is < median
when the curve is
positively skewed to right
 The mean is > media
when the curve is
negatively skewed to left
 The mean, median and mode are equal when distribution
is symmetrical.
 The mean is equal to median when it is symmetrical
Measures of Dispersion (Variation)
(Indicate spread of value)
The observations whether homogenous or
heterogeneous, the variability of the observations
1.
2.
3.
4.
5.
6.
Range
Variance
Standard deviation
Coefficient of variation
Standard error
Percentiles & quartiles
Describing Variability: the Range

Simplest & most obvious way of describing variability
Range = Highest - Lowest
 The range only takes into account the two extreme
scores and ignores any values in between.
 To counter this there the distribution is divided into
quarters (quartiles). Q1 = 25%, Q2 =50%, Q3 =75%
 The Interquartile range: the distance of the middle
two quartiles (Q3 – Q1)
 The Semi-Interquartile range: is one half of the
Interquartile range
Measures of Dispersion (Variation)
(Indicate spread of value)
The observations whether homogenous or
heterogeneous, the variability of the observations

1.

Range
The range is the difference between the largest
and the smallest observations.

Range = maximum – minimum

Disadvantage: it depends only on two values &
doesn’t take into account other observations
Measures of Dispersion (Variation)
(Indicate spread of value)
2.
Variance

It measures the spread of the observations around
the mean.

If the observations are close to their mean, the
variance is small, otherwise the variance is large.
Variance = S2 =

 (x
i

 x)
n 1
2
Describing Variability: Deviation

A more sophisticated measure of variability is one that
shows how scores cluster around the mean
 Deviation is the distance of a score from the mean
X - , e.g. 11 - 6.35 = 3.65, 3 – 6.35 = -3.35
 A measure representative of the variability of all the
scores would be the mean of the deviation scores
(X - )
Add all the deviations and divide by n
n
 However the deviation scores add up to zero (as
mean serves as balance point for scores)
Describing Variability: Variance
X
3
3
4
4
4
5
5
5
6
6
6
6
7
7
8
8
9
10
10
11
Sum
X-
-3.35
-3.35
-2.35
-2.35
-2.35
-1.35
-1.35
-1.35
-0.35
-0.35
-0.35
-0.35
0.65
0.65
1.65
1.65
2.65
3.65
3.65
4.65
0
(X -)²
11.22
11.22
5.52
5.52
5.52
1.82
1.82
1.82
0.12
0.12
0.12
0.12
0.42
0.42
2.72
2.72
7.02
13.32
13.32
21.62
106.55

To remove the +/- signs we
simply square each deviation
before finding the average. This
is called the Variance:

(X - )²
n

The numerator is referred to as
the Sum of Squares (SS): as it
refers to the sum of the squared
deviations around the mean
value
= 106.55
20
=
5.33
Describing Variability: Population Variance

Population variance is designated by ²
² = (X - )² = SS
N
N

Sample Variance is designated by s²
 Samples are less variable than populations: they
therefore give biased estimates of population
variability
 Degrees of Freedom (df): the number of independent
(free to vary) scores. In a sample, the sample mean
must be known before the variance can be
calculated, therefore the final score is dependent on
earlier scores: df = n -1
s² = (X - M)² = SS = 106.55 = 5.61
n-1
n -1
20 -1
Describing Variability: the Standard Deviation
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Variance is a measure based on squared distances
In order to get around this, we can take the square root of
the variance, which gives us the standard deviation
Population () & Sample (s) standard deviation
 = (X - )²
N
s = (X - M)²
n-1
So for our memory score
example we simple take the
square root of the variance:
= 5.61 = 2.37
Measures of Dispersion (Variation)
3.
Standard deviation (SD)

It is the square root of the variance S = Both variance & SD
are measures of variation in a set of data. The larger they
are the more heterogeneous the distribution. SD is more
preferred than other measures of variation.

Usually about 70% of the observations lie within one SD of
their mean and about 95% lie within two SD of the mean

If we add or subtract a constant from all observations, the
changed by the same constant, but the SD does not change
If we multiply or divide all the observation by the same
constant, both mean & SD changed by the same amount
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Small SD, the bell is tall & narrow
Large SD, the bell is short & broad
Standard Deviation (SD)
Example: Calculate SD for this observation set: (7,3,4,6)
Value
Xi
Deviation from mean
(Xi – X)
(Deviation)2
(Xi – X)2
7
2
4
3
-2
4
4
-1
1
6
1
1
20
0
10
Mean (X) = 20 = 5
4
SD =
2.5
Mean of (Dev.)2
=
1.6
= 10 = 2.5
4
Measures of Dispersion (Variation)
4.

Coefficient of variation
C.V expresses the SD as a percentage of the sample mean
s
x

C.V =
* 100

C.V = It is used to compare the relative variation of
uncorrelated quantities (blood glucose & cholesterol level)
Measures of Dispersion (Variation)

5. Standard error
SE measures how precisely the pp mean is estimated by
sample mean. The size of SE depends both on how much
variation there is in the pp and on the size of the sample.
s
n

SE =

SE = If the SE is large, sample is not precise to estimate
the pp.
Describing Variability


Describes in an exact quantitative measure, how
spread out/clustered together the scores are
Variability is usually defined in terms of distance
 How far apart scores are from each other
 How far apart scores are from the mean
 How representative a score is of the data set as a
whole
Quartiles & Interquartiles
 The age range of this group of 18 students is 55 – 25 = 30 years
 If the older student was not present, the range would have been 45 – 25
= 20 years
 This means that a single value could give non-real wide range of the
groups age
 Since we can not ignore a single value and we do not want to give
wrong impression, we estimate the interquartile range
25 28 30 30 32 33 34 35 36 37 40 40 41 42 44 45 45 55
Quartiles & Interquartiles
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1
The values are arranged in ascending manner
The groups then divided into 4 equal parts, each part contain one
quarter of observations
In the below example, 18/4 = 4.5 individuals
The value of the fifth individual is the minimum value of the
interquartile range
As a general rule, when the product of division contains a fraction then
take the following individual’s value (4.5, take the value of the fifth)
Interquartile range = 42 – 32 = 10 years
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18
25 28 30 30 32 33 34 35 36 37 40 40 41 42 44 45 45 55
First quartiles
Second quartile Third quartile
Interquartile range
Fourth quartile
Percentiles
 Used when the number of observations is large
 The values are arranged in ascending manner
 When the individuals are hundred, the lowest value
will be 1st percentile and the highest will be the
100th percentiles.