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WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
WFM 5201: Data Management and Statistical Analysis
Akm Saiful Islam
Lecture-1: Descriptive Statistics[Measures of central tendency]
April, 2008
Institute of Water and Flood Management (IWFM)Bangladesh University of Engineering and Technology (BUET)
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Descriptive Statistics
Measures of Central Tendency Measures of Location Measures of Dispersion Measures of Symmetry Measures of Peakdness
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Measures of Central Tendency
The central tendency is measured by averages. These describe the point about which the various observed values cluster.
In mathematics, an average, or central tendency of a data set refers to a measure of the "middle" or "expected" value of the data set.
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Measures of Central Tendency
Arithmetic Mean Geometric Mean Weighted Mean Harmonic Mean Median Mode
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Arithmetic Mean The arithmetic mean is the sum of a set of
observations, positive, negative or zero, divided by the number of observations. If we have “n” real numbers
their arithmetic mean, denoted by , can be expressed as:
n
xxxxx n
.............321
n
xx
n
ii
1
,.......,,,, 321 nxxxx
x
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Arithmetic Mean of Group Data
if are the mid-values and
are the corresponding frequencies, where the subscript ‘k’ stands for the number of classes, then the mean is
i
ii
f
zfz
kzzzz .,,.........,, 321
kffff ,........,,, 321
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Geometric Mean
Geometric mean is defined as the positive root of the product of observations. Symbolically,
It is also often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as data on the growth of the human population or interest rates of a financial investment.
Find geometric mean of rate of growth: 34, 27, 45, 55, 22, 34
nnxxxxG /1
321 )(
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Geometric mean of Group data
If the “n” non-zero and positive variate-values occur times, respectively, then the geometric mean of the set of observations is defined by:
Nn
i
fi
Nfn
ff in xxxxG
1
1
1
2121
n
iifN
1Where
nxxx ,........,, 21 nfff ,.......,, 21
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Geometric Mean (Revised Eqn.)
)( 321 nxxxxG
n
iixLog
NAntiLogG
1
1
n
iii xLogf
NAntiLogG
1
1
)( 321321 nfff xxxxG
Ungroup Data Group Data
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Harmonic Mean
Harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of average.
Typically, it is appropriate for situations when the average of rates is desired. The harmonic mean is the number of variables divided by the sum of the reciprocals of the variables. Useful for ratios such as speed (=distance/time) etc.
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Harmonic Mean Group Data
The harmonic mean H of the positive real numbers x1,x2, ..., xn is defined to be
n
i i
i
x
f
nH
1
n
i ix
nH
1
1
Ungroup Data Group Data
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Exercise-1: Find the Arithmetic , Geometric and Harmonic Mean Class Frequency
(f)x fx f Log x f / x
20-29 3 24.5 73.5 4.17 8.17
30-39 5 34.5 172.5 7.69 6.9
40-49 20 44.5 890 32.97 2.23
50-59 10 54.5 545 17.37 5.45
60-69 5 64.5 322.5 9.05 12.9
Sum N=43 2003.5 71.24 35.64
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Weighted Mean
The Weighted mean of the positive real numbers
x1,x2, ..., xn with their weight w1,w2, ..., wn is
defined to be
n
ii
n
iii
w
xw
x
1
1
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Median
The implication of this definition is that a median is the middle value of the observations such that the number of observations above it is equal to the number of observations below it.
)1(2
1
n
e XM
1222
1nne XXM
If “n” is odd If “n” is Even
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Median of Group Data
L0 = Lower class boundary of the median class h = Width of the median class f0 = Frequency of the median class F = Cumulative frequency of the pre- median class
Fn
f
hLM
ooe 2
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Steps to find Median of group data
1. Compute the less than type cumulative frequencies.
2. Determine N/2 , one-half of the total number of cases.
3. Locate the median class for which the cumulative frequency is more than N/2 .
4. Determine the lower limit of the median class. This is L0.
5. Sum the frequencies of all classes prior to the median class. This is F.
6. Determine the frequency of the median class. This is f0.
7. Determine the class width of the median class. This is h.
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Example-3:Find Median
Age in years Number of births Cumulative number of births
14.5-19.5 677 677
19.5-24.5 1908 2585
24.5-29.5 1737 4332
29.5-34.5 1040 5362
34.5-39.5 294 5656
39.5-44.5 91 5747
44.5-49.5 16 5763
All ages 5763 -
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Mode
Mode is the value of a distribution for which the frequency is maximum. In other words, mode is the value of a variable, which occurs with the highest frequency.
So the mode of the list (1, 2, 2, 3, 3, 3, 4) is 3. The mode is not necessarily well defined. The list (1, 2, 2, 3, 3, 5) has the two modes 2 and 3.
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Example-2: Find Mean, Median and Mode of Ungroup Data
The weekly pocket money for 9 first year pupils was found to be:
3 , 12 , 4 , 6 , 1 , 4 , 2 , 5 , 8
Mean5
Mode4
Median4
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Mode of Group Data
L1 = Lower boundary of modal class
Δ1 = difference of frequency between
modal class and class before it Δ2 = difference of frequency between
modal class and class after H = class interval
hLM21
110
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Steps of Finding Mode
Find the modal class which has highest frequency
L0 = Lower class boundary of modal class
h = Interval of modal class
Δ1 = difference of frequency of modal
class and class before modal class
Δ2 = difference of frequency of modal class and
class after modal class
WFM 5201: Data Management and Statistical Analysis © Dr. Akm Saiful Islam
Example -4: Find Mode
Slope Angle (°)
Midpoint (x) Frequency (f) Midpoint x
frequency (fx)
0-4 2 6 12
5-9 7 12 84
10-14 12 7 84
15-19 17 5 85
20-24 22 0 0
Total n = 30 ∑(fx) = 265