Skewness & Kurtosis: Reference
Source: http://mathworld.wolfram.com/NormalDistribution.html
Further Moments – Skewness
• Skewness measures the degree of asymmetry
exhibited by the data
n
skewness 
 (x  x)
i 1
3
i
ns
3
• If skewness equals zero, the histogram is
symmetric about the mean
• Positive skewness vs negative skewness
• Skewness measured in this way is sometimes
referred to as “Fisher’s skewness”
Further Moments – Skewness
Source: http://library.thinkquest.org/10030/3smodsas.htm
n
Mode
Median
skewness 
Mean
A
B
3
(
x

x
)
 i
i 1
ns
3
Median
Mean
n = 26 mean = 4.23 median = 3.5 mode = 8
n
skewness 
Value
1
2
3
4
5
6
7
8
9
10
Occurrences
1
4
8
4
3
2
1
1
1
1
Deviation
(1 – 4.23) = -3.23
(2 – 4.23) = -2.23
(3 – 4.23) = -1.23
(4 – 4.23) = -0.23
(5 – 4.23) = 0.77
(6 – 4.23) = 1.77
(7 – 4.23) = 2.77
(8 – 4.23) = 3.77
(9 – 4.23) = 4.77
(10 - 4.23)= 5.77
Mean = 4.23
s = 2.27
3
(
x

x
)
 i
i 1
ns
Cubed deviation
3
Occur*Cubed
(-3.23)3 = -33.70
(-2.23)3 = -11.09
(-1.13)3 = -1.86
(-0.23)3 = -0.01
(+0.77)3 = 0.46
(+1.77)3 = 5.54
(+2.77)3 = 21.25
(+3.77)3 = 53.58
(+4.77)3 = 108.53
(+5.77)3 = 192.10
-33.70
-44.36
-14.89
-0.05
1.37
11.09
21.25
53.58
108.53
192.10
Sum = 294.94
Skewness = 0.97
n
Mode
Median
Mean
skewness 
 (x  x)
i
i 1
ns
Skewness > 0 (Positively skewed)
3
3
n
skewness 
3
(
x

x
)
 i
Mode
i 1
ns 3
Median
Mean
A
B
Skewness < 0 (Negatively skewed)
n
skewness 
 (x  x)
i 1
i
ns 3
Source: http://mathworld.wolfram.com/NormalDistribution.html
Skewness = 0 (symmetric distribution)
3
Skewness – Review
• Positive skewness
– There are more observations below the mean
than above it
– When the mean is greater than the median
• Negative skewness
– There are a small number of low observations
and a large number of high ones
– When the median is greater than the mean
Kurtosis – Review
• Kurtosis measures how peaked the histogram is
(Karl Pearson, 1905)
n
kurtosis 
 (x  x)
i
i
ns
4
4
3
• The kurtosis of a normal distribution is 0
• Kurtosis characterizes the relative peakedness
or flatness of a distribution compared to the
normal distribution
Kurtosis – Review
• Platykurtic– When the kurtosis < 0, the
frequencies throughout the curve are closer to be
equal (i.e., the curve is more flat and wide)
• Thus, negative kurtosis indicates a relatively flat
distribution
• Leptokurtic– When the kurtosis > 0, there are
high frequencies in only a small part of the curve
(i.e, the curve is more peaked)
• Thus, positive kurtosis indicates a relatively
peaked distribution
n
kurtosis 
 (x  x)
i
i
ns
4
4
3
Source: http://espse.ed.psu.edu/Statistics/Chapters/Chapter3/Chap3.html
Measures of central tendency –
Review
• Measures of the location of the middle or
the center of a distribution
• Mean
• Median
• Mode
Mean – Review
• Mean – Average value of a distribution; Most
commonly used measure of central tendency
• Median – This is the value of a variable such
that half of the observations are above and half
are below this value, i.e., this value divides the
distribution into two groups of equal size
• Mode - This is the most frequently occurring
value in the distribution
An Example Data Set
• Daily low temperatures recorded in Chapel Hill
(01/18-01/31, 2005, °F)
Jan. 18 – 11
Jan. 19 – 11
Jan. 20 – 25
Jan. 21 – 29
Jan. 22 – 27
Jan. 23 – 14
Jan. 24 – 11
Jan. 25 – 25
Jan. 26 – 33
Jan. 27 – 22
Jan. 28 – 18
Jan. 29 – 19
Jan. 30 – 30
Jan. 31 – 27
• For these 14 values, we will calculate all three
measures of central tendency - the mean, median,
and mode
Mean – Review
• Mean –Most commonly used measure of central
tendency
• Procedures
• (1) Sum all the values in the data set
• (2) Divide the sum by the number of values in the
n
data set
• Watch for outliers
x
x
i 1
n
i
Mean – Review
• (1) Sum all the values in the data set
 11 + 11 + 11 + 14 + 18 + 19 + 22 + 25 + 25 + 27 +
27 + 29 + 30 + 33 = 302
n
• (2) Divide the sum by the number
of values in the data set
 Mean = 302/14 = 21.57
x
x
i 1
i
n
• Is this a good measure of central tendency for this
data set?
Median – Review
• Median - 1/2 of the values are above it & 1/2 below
• (1) Sort the data in ascending order
• (2) Find the value with an equal number of values
above and below it
• (3) Odd number of observations  [(n-1)/2]+1 value
from the lowest
• (4) Even number of observations  average (n/2)
and [(n/2)+1] values
• (5) Use the median with asymmetric distributions,
particularly with outliers
Median – Review
• (1) Sort the data in ascending order:
 11, 11, 11, 14, 18, 19, 22, 25, 25, 27, 27, 29, 30, 33
• (2) Find the value with an equal number of values
above and below it
Even number of observations  average the
(n/2) and [(n/2)+1] values
 (14/2) = 7; [(14/2)+1] = 8
 (22+25)/2 = 23.5 (°F)
• Is this a good measure of central tendency for this
data?
Mode – Review
• Mode – This is the most frequently occurring value
in the distribution
• (1) Sort the data in ascending order
• (2) Count the instances of each value
• (3) Find the value that has the most occurrences
• If more than one value occurs an equal number of
times and these exceed all other counts, we have
multiple modes
• Use the mode for multi-modal data
Mode – Review
• (1) Sort the data in ascending order:
 11, 11, 11, 14, 18, 19, 22, 25, 25, 27, 27, 29, 30, 33
• (2) Count the instances of each value:
 11, 11, 11, 14, 18, 19, 22, 25, 25, 27, 27, 29, 30, 33
3x
1x 1x 1x 1x
2x
2x
1x 1x 1x
• (3) Find the value that has the most occurrences
 mode = 11 (°F)
• Is this a good measure of the central tendency of
this data set?
Measures of Dispersion – Review
• In addition to measures of central tendency,
we can also summarize data by characterizing its
variability
• Measures of dispersion are concerned with the
distribution of values around the mean in data:
– Range
– Interquartile range
– Variance
– Standard deviation
– z-scores
– Coefficient of Variation (CV)
An Example Data Set
• Daily low temperatures recorded in Chapel Hill
(01/18-01/31, 2005, °F)
Jan. 18 – 11
Jan. 19 – 11
Jan. 20 – 25
Jan. 21 – 29
Jan. 22 – 27
Jan. 23 – 14
Jan. 24 – 11
Jan. 25 – 25
Jan. 26 – 33
Jan. 27 – 22
Jan. 28 – 18
Jan. 29 – 19
Jan. 30 – 30
Jan. 31 – 27
• For these 14 values, we will calculate all measures
of dispersion
Range – Review
• Range – The difference between the largest and
the smallest values
• (1) Sort the data in ascending order
• (2) Find the largest value
 max
• (3) Find the smallest value
 min
• (4) Calculate the range
 range = max - min
• Vulnerable to the influence of outliers
Range – Review
• Range – The difference between the largest and
the smallest values
• (1) Sort the data in ascending order
 11, 11, 11, 14, 18, 19, 22, 25, 25, 27, 27, 29, 30, 33
• (2) Find the largest value
 max = 33
• (3) Find the smallest value
 min = 11
• (4) Calculate the range
 range = 33 – 11 = 22
Interquartile Range – Review
• Interquartile range – The difference between the
25th and 75th percentiles
• (1) Sort the data in ascending order
• (2) Find the 25th percentile – (n+1)/4 observation
• (3) Find the 75th percentile – 3(n+1)/4 observation
• (4) Interquartile range is the difference between
these two percentiles
Interquartile Range – Review
• (1) Sort the data in ascending order
 11, 11, 11, 14, 18, 19, 22, 25, 25, 27, 27, 29, 30, 33
• (2) Find the 25th percentile – (n+1)/4 observation
 (14+1)/4 = 3.75
 11+(14-11)*0.75 = 13.265
• (3) Find the 75th percentile – 3(n+1)/4 observation
 3(14+1)/4 = 11.25  27+(29-27)*0.25 = 27.5
• (4) Interquartile range is the difference between
these two percentiles
 27.5 – 13.265 = 14.235
Variance – Review
• Variance is formulated as the sum of squares of
statistical distances (or deviation) divided by
the population size or the sample size minus one:
n
s 
2
 (x  x)
i 1
i
n 1
2
Variance – Review
• (1) Calculate the mean

x
• (2) Calculate the deviation for each value

xi  x
• (3) Square each of the deviations

( xi  x )
2
• (4) Sum the squared deviations

2
(
x

x
)
 i
• (5) Divide the sum of squares by (n-1) for a sample

2
(
x

x
)
/( n  1)
 i
Variance – Review
• (1) Calculate the mean

x  25.7
• (2) Calculate the deviation for each value

xi  x
Jan. 18 (11 – 25.7) = -10.57
Jan. 25 (25 – 25.7) = 3.43
Jan. 19 (11 – 25.7) = -10.57
Jan. 26 (33 – 25.7) = 11.43
Jan. 20 (25 – 25.7) = 3.43
Jan. 27 (22 – 25.7) = 0.43
Jan. 21 (29 – 25.7) = 7.43
Jan. 28 (18 – 25.7) = -3.57
Jan. 22 (27 – 25.7) = 5.43
Jan. 29 (19 – 25.7) = -2.57
Jan. 23 (14 – 25.7) = -7.57
Jan. 30 (30 – 25.7) = 8.42
Jan. 24 (11 – 25.7) = -10.57
Jan. 31 (27 – 25.7) = 5.42
Variance – Review
• (3) Square each of the deviations

( xi  x )
2
Jan. 18 (-10.57)^2 = 111.76
Jan. 25 (3.43)^2 = 11.76
Jan. 19 (-10.57)^2 = 111.76
Jan. 26 (11.43)^2 = 130.61
Jan. 20 (3.43)^2 = 11.76
Jan. 27 (0.43)^2 = 0.18
Jan. 21 (7.43)^2 = 55.18
Jan. 28 (-3.57)^2 = 12.76
Jan. 22 (5.43)^2 = 29.57
Jan. 29 (-2.57)^2 = 6.61
Jan. 23 (7.57)^2 = 57.33
Jan. 30 (8.43)^2 = 71.04
Jan. 24 (-10.57)^2 = 111.76
Jan. 31 (5.43)^2 = 29.57
• (4) Sum the squared deviations

 (x  x)
i
2
= 751.43
Variance – Review
• (5) Divide the sum of squares by (n-1) for a
sample

2
(
x

x
)
/( n  1)
 i
= 751.43 / (14-1) = 57.8
• The variance of the Tmin data set (Chapel Hill)
is 57.8
Standard Deviation – Review
• Standard deviation is equal to the square root
of the variance
n
s
 (x  x)
i 1
2
i
n 1
• Compared with variance, standard deviation
has a scale closer to that used for the mean and
the original data
Standard Deviation – Review
• (1) Calculate the mean

x
• (2) Calculate the deviation for each value

xi  x
• (3) Square each of the deviations

( xi  x )
2
• (4) Sum the squared deviations

2
(
x

x
)
 i
• (5) Divide the sum of squares by (n-1) for a sample

(x  x)
i
2
/( n  1)
• (6) Take the square root of the resulting variance

2
(
x

x
)
/( n  1)
 i
Standard Deviation – Review
• (1) – (5)
 s2 = 57.8
• (6) Take the square root of the variance

57.8  7.6
• The standard deviation (s) of the Tmin
data set (Chapel Hill) is 7.6 (°F)
z-score – Review
• Since data come from distributions with different
means and difference degrees of variability, it is
common to standardize observations
• One way to do this is to transform each observation
into a z-score
xi  x
z
s
• May be interpreted as the number of standard
deviations an observation is away from the mean
z-scores – Review
• z-score is the number of standard deviations an
observation is away from the mean
• (1) Calculate the mean

x
• (2) Calculate the deviation

xi  x
• (3) Calculate the standard deviation
s

2
(
x

x
)
/( n  1)
 i
• (4) Divide the deviation by standard deviation

z  ( xi  x ) / s
z-scores – Review
• Z-score for maximum Tmin value (33 °F)
• (1) Calculate the mean

x  21.57
• (2) Calculate the deviation

xi  x  11.43
• (3) Calculate the standard deviation (SD)

2
(
x

x
)
/( n  1)  7.6
 i
• (4) Divide the deviation by standard deviation

z  ( xi  x ) / s  11.43 / 7.6  1.50
Coefficient of Variation – Review
• Coefficient of variation (CV) measures the spread
of a set of data as a proportion of its mean.
• It is the ratio of the sample standard deviation to
the sample mean
s
CV   100%
x
• It is sometimes expressed as a percentage
• There is an equivalent definition for the coefficient
of variation of a population
Coefficient of Variation – Review
• (1) Calculate mean

x
• (2) Calculate standard deviation

s
2
(
x

x
)
/( n  1)
 i
• (3) Divide standard deviation by mean

CV =
s
100%
x
Coefficient of Variation – Review
• (1) Calculate mean

x  25.7
• (2) Calculate standard deviation

s
 (x  x)
i
2
/( n  1)  7.6
• (3) Divide standard deviation by mean
 CV = s  100%  7.6 / 25.7  100%  29.58
x
Histograms – Review
• We may also summarize our data by constructing
histograms, which are vertical bar graphs
• A histogram is used to graphically summarize
the distribution of a data set
• A histogram divides the range of values in a data
set into intervals
• Over each interval is placed a bar whose height
represents the percentage of data values in the
interval.
Building a Histogram – Review
• (1) Develop an ungrouped frequency table
 11, 11, 11, 14, 18, 19, 22, 25, 25, 27, 27, 29, 30, 33

11
3
14
1
18
1
19
1
22
1
25
2
27
2
29
1
30
1
33
1
Building a Histogram – Review
• 2. Construct a grouped frequency table
 Select a set of classes

11-15
4
16-20
2
21-25
3
26-30
4
31-35
1
Building a Histogram – Review
• 3. Plot the frequencies of each class
Box Plots – Review
• We can also use a box plot to graphically
summarize a data set
• A box plot represents a graphical summary of
what is sometimes called a “five-number
summary” of the distribution
– Minimum
– Maximum
– 25th percentile
– 75th percentile
– Median
• Interquartile Range (IQR)
max.
median
min.
Rogerson, p. 8.
75th
%-ile
25th
%-ile
Boxplot – Review
Further Moments of the Distribution
• While measures of dispersion are useful for helping
us describe the width of the distribution, they tell us
nothing about the shape of the distribution
Source: Earickson, RJ, and Harlin, JM. 1994. Geographic Measurement and Quantitative Analysis. USA:
Macmillan College Publishing Co., p. 91.
Skewness – Review
• Skewness measures the degree of asymmetry
exhibited by the data
• Positive skewness – More observations below
the mean than above it
• Negative skewness – A small number of low
observations and a large number of high ones
n
skewness 
 (x  x)
i 1
i
ns
3
3
For the example data set:
Skewness = -0.1851
Skewness = -0.1851 (Negatively skewed)
Kurtosis – Review
• Kurtosis measures how peaked the histogram is
• Leptokurtic: a high degree of peakedness
– Values of kurtosis over 0
• Platykurtic: flat histograms
– Values of kurtosis less than 0
n
kurtosis 
 (x  x)
i
i
ns
4
4
3
For the example data set:
Kurtosis = -1.54 < 0
Kurtosis = -1.54 < 0 (Platykurtic)