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Data Analysis Techniques II:
Measures of Central Tendencies,
Dispersion and Symmetry
Advanced Planning Techniques, Lecture 9
Prof. Dr. S. Shabih-ul-Hassan Zaidi
Dean, Faculty of Architecture & Planning,
University of Engineering & Technology, Lahore.
The Mean
The Arithmetic Mean or Average can be defined as the sum of
the value of all items divided by the number of items. The
mean is a very important measure of the central tendency of
the data, because it not only gives a summary of the data but
is also used in further analysis. In case of grouped data the
Mean is calculated by the formula:
Mean = X = Sum f.x/ n
Where f = Frequency of the group
x = Central value of the group i.e. lower limit + upper limit
divided by 2
n = Total of the frequency
The Median
The Median is the central item of a series. Its value divides the
series into two parts. In case of even number of items, the
average of the two central items is taken as Median. In case of
grouped data the median can be calculated by the formula:
Median = l + i/f (n/2 – c)
Where l = lower limit of the median group i.e the group in
which the value of n/2 lies
i = class interval of the median group
f = frequency of the median group
n = total of the frequency
c = cumulative frequency of the group preceding the
median group
The Quartiles, Deciles and the
Percentiles
The median divides the series into two parts. Similarly, the
series can be divided into 4 parts and the value of quartiles can
be calculated. On the same pattern, the value of Deciles (10th
part) and the value of Percentiles (100th part) can be
calculated. In case of grouped data the formulas for calculation
of these measures are given below:
First Quartile = l + i/f (n/4 – c)
Third Quartile = l + i/f (3n/4 – c)
First Deciles = l + i/f (n/10 – c)
Ninth Decile = l + i/f (9n/10 – c)
First Percentile = l + i/f (n/100 - c)
Ninety ninth Percentile = l + i/f (99n/100 – c)
The Mode
The mode represents the value of the item which is
repeated the maximum number of times in a series. In
case of grouped data the mode can be calculated by
the following formula:
Mode = l1 + f2/(f1 + f2)xi
Where l1 = Lower limit of the modal group i.e.
maximum frequency group
f1 = Frequency of the modal group
f2 = Frequency of the next higher frequency
group
i = Class interval
Measures of Dispersion
The measures of dispersion tell about the dispersion or scatter of the data.
The following measures of dispersion are usually calculated:
Range: Max. value – Min. value, or
Range = Fn – F1
Mean Deviation = Sum |x – x-|/n
Where x is the central value of the group
x- = Mean n = Total of frequencies
Variance = Sum f (x – x-)2/n
Standard Deviation = Under-root of Variance
Quartile Deviation = (Q3 – Q1 ) / 2
Measure of Symmetry
A perfectly symmetrical data can be represented by a bell
shaped curve. In this case the value of mean, median and the
mode is the same. The opposite of symmetry is skewness.
Therefore, the Pearson’s Coefficient of Skewness is calculated
for measuring symmetry of the data. The formula is:
Coefficient of Skewness = (Mean – Mode)/Standard Deviation
or
Coeff. of Skewness = 3(Mean - Median)/Standard Deviation
If the value of Coefficient is negative, the distribution is said to
be negatively skewed i.e. when large number of items have
small values, while the distribution is said to be positively
skewed when the a large number of items have large values.
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