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Descriptive Statistics
Anwar Ahmad
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
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
Arithmetic Mean
Example:
4,5,9,1,2
21/5
4.2
1
1
x  x1  x 2    xn  
n
n
n
x
i
i 1
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
Mode
• The most frequently occurring value in a
set of observations.1,2,2,4,5,9
• 2 is the mode
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
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
i
1
xin
Harmonic Mean
• The harmonic mean of a set of
n
observations is the reciprocal
(1/x) of the arithmetic mean of
the reciprocals of the
observations.
1
i xi
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
Harmonic Mean
• Harmonic mean
• 2/ (1/30+1/60) = 40
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%
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
Measure of Variability
• Range
• Variance
• Standard Deviation
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.
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
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
n 1
i
xi
x2
Variance
• 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
• 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.
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
n 1
i
xi x 2
Summary Formulae