GUIDELINES FOR CONSTRUCTING FREQUNCY DISTRIBUTIONS:
The purpose of constructing frequency distributions is to condense a body of data into a table that is readily appraised and comprehended.
Definition of class intervals:
Each observation in the data set must belong to one and only one class interval.
Width of class intervals:
Class intervals of equal widths, if possible. There are situations where unequal widths are necessary for various reasons, e.g., income distribution.
Number of Class Intervals:
No hard-and-fast rules, but generally between 5 and 20.
Relationship between Number and Width of Class Intervals:
Width of class interval = (largest value – smallest value)/ No. of class intervals.
Position of Class Intervals:
Midpoints should be close to the average of the observations included in the class interval.
THE CUMULATIVE FREQUENCY DISTRIBUTION:
It shows the number of measurements in the population that lie below or above a certain value.
e.g., how many firms in Table 1.8 paid their chief executives less than $1.4 million in 1991?
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Salary
No. of firms (millions $)
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Less than 0.20
0
Less than 0.60
5
Less than 1.00
8
Less than 1.40
16
Table 1.11:
Less than 1.80
20
Cumulative
Less than 2.20
23
Frequency
Less than 2.60
24
Distriution
Less than 3.00
24
Less than 3.40
25
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One can plot the cumulative frequency distributions, called OGIVE. All these are less-than cumulative frequency distributions.
One can also construct greater-than cumulative frequency distributions and ogives, which show the number of frequencies that exceed particular values.
FREQUENCY DISTRIBUTIONS FOR SAMPLE DATA:
Sample vs. Entire population
Generally we do not have the frequency distribution of the entire
population. We have frequency distribution based on a sample. They will
be somewhat different. It is good to visualize the frequency distribution
of the population, and how this is related to that based on only one sample.
SUMMARY MEASURES:
Measures of Central tendency and Dispersion:
Central tendency: Þ average level, typical value.
Depending on the purpose, there are a number of alternative measures of central tendency.
Dispersion: Þ degree to which the individual measurements vary about the average value.
Think of a society with two individuals. In one their yearly incomes are $50,000 each. In the other, one earns $100,000 and the other $0. In both societies the mean income is $50,000. However, the dispersion is much higher in the latter society.
PARAMETERS and STATISTICS:
Population parameters are seldom known. We are interested in drawing inference about population parameters from a sample.
Estimates of population parameters based on a sample is known as statistics.
Arithmetic mean: Sum of the numbers in the sample divided by their number.
m = (X1 + X2 + …. + XN)/N
m is the population value.
When based on a sample
X = (X1 + X2 + … + Xn)/n
= ( S Xi )/n
Calculation of arithmetic mean from grouped data:
Use the midpoint of the class interval as the value of measurements in that class as an approximation.
Use table 2.2:
X = S (f.i XI)/n