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Descriptive Statistics
Anwar Ahmad
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Central Tendency- Measure of location
• Measures descriptive of a typical or representative value in a group of observations
• It applies to groups rather than individuals
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Arithmetic Mean
• Simplest and obvious measure of central tendency
• Simple average of the observations in the group, i.e. the value obtained by adding the observations together and dividing this sum by the number of observations in the group
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Arithmetic Mean
Example:
4,5,9,1,2
21/5
4.2
n
i
in xn
xxxn
x1
1121
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Median
• The middle value in a set of observations ordered by size
• Median income or median house price
• 1,2,4,5,9
• 4 is the median
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Mode
• The most frequently occurring value in a set of observations.1,2,2,4,5,9
• 2 is the mode
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Other Measures of Central Tendency
• Midrange: The value midway between the smallest and largest values in the sample, that is, the arithmetic mean the largest and smallest values, the extremes.
• 4,5,9,1,2
• (9+1)/2
• 5
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Geometric Mean
• The geometric mean of a set of observations is the nth root of their product.
• Gm of 4 & 9• Sqrt 4*9• Sqrt 36• 6
ixi1n
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Harmonic Mean
• The harmonic mean of a set of observations is the reciprocal
(1/x) of the arithmetic mean of the reciprocals of the
observations.
ni
1
xi
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Harmonic Mean
• Av. Velocity of car that traveled first 10 mi. at 30 mph; and the second 10 mi. at 60 mph.
• Mean 30+60 /2 = 45 ?
• Total distance by total time
• 10+10 / 1/3 + 1/6 hr (1/2 hr)
• 20/ ½ hr
• Av. velocity 40 mph
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Harmonic Mean
• Harmonic mean
• 2/ (1/30+1/60) = 40
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Weighted Mean
• When all observations do not have equal weight
• Lab A 50 cultures, 25 positive, 50%• Lab B 80 cultures, 60 positive, 75%• Lab C 120 cultures, 30 positive, 25% =
150/3 =50%• WM = 50(50%)+80(75%)+120(25%) /
50+80+12• 46%
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Measure of Variability
• 1,4,4,4,7 = 20 = 20/5 = 4 variation
• 4,4,4,4,4 = 20 = 20/5 = 4 no variation
• Same means, median, mode
• 0 if no variation
• Some + value, if there is a variation
• Variation from the mean
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Measure of Variability
• Range
• Variance
• Standard Deviation
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Range • Range is the simplest measure of spread or
dispersion: • It is the difference between the largest and
the smallest values. • The range can be a useful measure of spread
because it is easily understood. • However, it is very sensitive to extreme
scores since it is based on only two values.
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Range
• The range should almost never be used as the only measure of spread, but can be informative if used as a supplement to other measures of spread such as the standard deviation or variance
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Variance
• Squared deviation from the mean.
• 1,4,4,4,7, mean 4• (1-4), (4-4), (4-4), (4-4), (7-4)• -3, 0, 0, 0, 3 = 0• -32, 0, 0, 0, 32 = 18/5 = 18/4 =
4.5
1
n1
ixi x2
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• The variance describes the heterogeneity of a distribution and is calculated from a formula that involves every score in the distribution. It is typically symbolized by the letter s with a superscript "2". The formula is
Variance, s2 = sum (scores - mean)2/(n - 1) degree of freedom
Variance
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Variance
• The variance is a measure of how spread out a distribution is. It is computed as the average squared deviation of each number from its mean.
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Standard deviation
• The square root (the positive one) of the variance is known as the "standard deviation." It is symbolized by s with no superscript.
• Sqrt 4.5• 2.12
1
n1
ixi x2
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Summary Formulae
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