- Level: AS and A Level
- Subject: Maths
- Word count: 1786
Frequency curves and frequency tables
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Introduction
Question 1
The usefulness of drawing a cumulative frequency curve varies.
When a relatively large number of observations or measurements has been made, it is useful to organize the data into classes according to the magnitude of the measurements. Grouping the data in classes makes the interpretation of the results easier and also provides the basis for portraying the results graphically, as demonstrated.
A frequency curve can be described as being smoothed frequency polygon. Thus, if we were to smooth the polygon, the result would be a frequency curve. To form a frequency curve can be described in two ways: in term of its departure from symmetry, which is called skewness, and in terms of its degree of peakedness, which is called kurtosis. A symmetrical frequency curve is one for which the right half of the curve is the mirror image of the left half of the curve. The concepts off skewness and kurtosis are important because they are used to describe probability curves, beginning with our description of probability distribution, as well as frequency curves.
In terms of skewness, a frequency curve can be
- negatively skewed . eg. Non-symmetrical with the longer ‘tail’ of the frequency to the left.
- Symmetrical
- Positively skewed. Eg.
Middle
10
154.5-159.5
11
159.5-164.5
6
164.5-169.5
4
169.5-174.5
2
174.5-179.5
1
Class width
The difference between the upper class boundary and the lower class boundary of the same class gives the class width.
The class width of the 1st class is (149.5-144.5), or 5 cm.
However, the data are grouped in classes, as in the above example, we only know that there are 6 students of heights in the interval (144.5-149.5) cm. We are unable to tell the actual height of each student, unless we refer to the original list of data.
In frequency tables there are three statistical averages, namely mode, median and mean.
Similarly we can also find the mode, median and mean grouped data.
Mode, Median, Mean
If the class (width) interval is the same for all classes, then
- the modal class refers to the class with the highest frequency.
- the median refers to the class in which the middles score lies.
- the mean refers to the average of the scores for the distribution.
Grouped data
For grouped data, we assume that mid-value of each class is taken to be the mean score for that class. Thus, mean for grouped data can be calculated as follows:
MEAN = ____Total fx___ , where x represents the mid-value of each class and f
Total frequency represents the corresponding frequency.
From the table as well, we can also find the quartile range. That is the lower quartile, upper quartile and interquartile range.
Lower quartile is value for which a quarter (25%) of the distribution which lie as below.
Upper quartile is the value for which three quarters (75%)of the distribution lie as below it.
Interquartile range is the difference between the upper and lower quartiles.
Example
The speeds in km/h, of 30 cars which traveled along an expressway were recorded as follows:
Speed (km/h) | 50-59 | 60-69 | 70-79 | 80-89 | 90-99 |
No. of cars | 3 | 5 | 7 | 15 | 1 |
Solution:
Speed (km/h) | Mid-value (x) | Frequency (fx) | Frequency Mid-value (fx) | Cumulative frequency |
50-59 | 54.5 | 2 | 109 | 2 |
60-69 | 64.5 | 5 | 322.5 | 7 |
70-79 | 74.5 | 7 | 521.5 | 14 |
80-89 | 84.5 | 15 | 1267.5 | 29 |
90-99 | 94.5 | 1 | 94.5 | 30 |
Total | - | 30 | 2315.0 | - |
Conclusion
Frequency curves
A cumulative frequency identifies the cumulative number of observations below the upper class boundary of each class is determined by adding the observed for that class to cumulative frequency for the preceding class. A smooth curve which corresponds to the limiting case of a histogram computed for a frequency distribution of a continuous distribution as the number of data points becomes very large.
It can also be potayed graphically by accumulative frequency polygon. This line graph is more frequently called as ogive, constructed by joining the mid-points of each interval with a straight line. This has been done on the diagram below. You may noticed it ‘finished off’ at each end, so that the area under the curves represents the total distributions.
REFERENCES
Ref. 1
Schaum’s Outline of Theory and problems
of Business Statistic
Fourth Edition Leonard J. Kazmier
W.P Carey School Of Business
Arizona State University
Ref. 2
Basic Statistics For Business and Economics
Second Edition
Leonard J. Kazmier
Norval F. Pohl
Arizona State University
Noethern Arizona University
Ref. 3
Even you can learn statistics
David M. Levine
David F. Stephan
@ 2003 Pearson Education, inc.
Publishing as Person Prentice Hall
Upper Saddle, NJ 07458
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