MEDIAN IN CASE OF A FREQUENCY DISTRIBUTION OF A CONTINUOUS VARIABLE
Statistics and ProbabilityMEDIAN IN CASE OF A FREQUENCY DISTRIBUTION OF A CONTINUOUS VARIABLE
In case of a frequency distribution, the median is given by the formula
~ h n
X =l + −c f 2
Where l =lower class boundary of the median class (i.e. that class for which the cumulative frequency is just in excess of n/2). h=class interval size of the median class f =frequency of the median class n=Σf (the total number of observations) c =cumulative frequency of the class preceding the median class
Note:
This formula is based on the assumption that the observations in each class are evenly distributed between the two class limits.
EXAMPLE:
Going back to the example of the EPA mileage ratings, we have
Mileage Rating | No. of Cars | Class Boundaries | Cumulative Frequency |
30.0 – 32.9 | 2 | 29.95 – 32.95 | 2 |
33.0 – 35.9 | 4 | 32.95 – 35.95 | 6 |
36.0 – 38.9 | 14 | 35.95 – 38.95 | 20 |
39.0 – 41.9 | 8 | 38.95 – 41.95 | 28 |
42.0 – 44.9 | 2 | 41.95 – 44.95 | 30 |
In this example, n = 30 and n/2 = 15. Thus the third class is the median class. The median lies somewhere between 35.95 and 38.95. Applying the above formula, we obtain
~3
X = 35.95 +(15 − 6)
14
= 35.95 +1.93 = 37.88 ~
− 37.9
INTERPRETATION
This result implies that half of the cars have mileage less than or up to 37.88 miles per gallon whereas the other half of the cars has mileage greater than 37.88 miles per gallon. As discussed earlier, the median is preferable to the arithmetic mean when there are a few very high or low figures in a series. It is also exceedingly valuable when one encounters a frequency distribution having openended class intervals. The concept of openended frequency distribution can be understood with the help of the following example.
Example:
WAGES OF WORKERS IN A FACTORY | |
Monthly Income (in Rupees) | No. of Workers |
Less than 2000/ | 100 |
2000/to 2999/ | 300 |
3000/to 3999/ | 500 |
4000/to 4999/ | 250 |
5000/and above | 50 |
Total | 1200 |
In this example, both the first class and the last class are openended classes. This is so because of the fact that we do not have exact figures to begin the first class or to end the last class. The advantage of computing the median in the case of an openended frequency distribution is that, except in the unlikely event of the median falling within an openended group occurring in the beginning of our frequency distribution, there is no need to estimate the upper or lower boundary. This is so because of the fact that, if the median is falling in an intermediate class, then, obviously, the first class is not being involved in its computation. The next concept that we will discuss is the empirical relation between the mean, median and the mode. This is a concept which is not based on a rigid mathematical formula; rather, it is based on observation. In fact, the word ‘empirical’ implies ‘based on observation’.
This concept relates to the relative positions of the mean, median and the mode in case of a humpshaped distribution. In a singlepeaked frequency distribution, the values of the mean, median and mode coincide if the frequency distribution is absolutely symmetrical.
But in the case of a skewed distribution, the mean, median and mode do not all lie on the same point. They are pulled apart from each other, and the empirical relation explains the way in which this happens. Experience tells us that in a unimodal curve of moderate skewness, the median is usually sandwiched between the mean and the mode.
The second point is that, in the case of many reallife datasets, it has been observed that the distance between the mode and the median is approximately double of the distance between the median and the mean, as shown below:
f
X
Mode
MedianMean
This diagrammatic picture is equivalent to the following algebraic expression: Median Mode 2 (Mean Median) (1) The abovementioned point can also be expressed in the following way:
Mean – Mode = 3 (Mean – Median) (2) Equation (1) as well as equation (2) yields the approximate relation given below:
EMPIRICAL RELATION BETWEEN THE MEAN, MEDIAN AND THE MODE
Mode = 3 Median – 2 Mean An exactly similar situation holds in case of a moderately negatively skewed distribution. An important point to note is that this empirical relation does not hold in case of a Jshaped or an extremely skewed distribution.
Let us now extend the concept of partitioning of the frequency distribution by taking up the concept of quantiles (i.e. quartiles, deciles and percentiles). We have already seen that the median divides the area under the frequency polygon into two equal halves:
Median
A further split to produce quarters, tenths or hundredths of the total area under the frequency polygon is equally possible, and may be extremely useful for analysis. (We are often interested in the highest 10% of some group of values or the middle 50% another.
QUARTILES
The quartiles, together with the median, achieve the division of the total area into four equal parts. The first, second and third quartiles are given by the formulae:
1. FIRST QUARTILE
h n
Q_{1 }= l + − c f 4
2. SECOND QUARTILE (I.E. MEDIAN)
h 2n h
Q_{2 }=l+−c =l+( 2−c)
f 4 f
3. THIRD QUARTILE
h 3 n
Q = l + − c ^{3 }f 4
It is clear from the formula of the second quartile that the second quartile is the same as the median.
Q1 Q2= X Q3
DECILES & PERCENTILES
The deciles and the percentiles give the division of the total area into 10 and 100 equal parts respectively. The formula for the first decile is
h n
D_{1 }= l + −c
f 10
The formulae for the subsequent deciles are
h 2n
D_{2 }= l + − c
f 10
D_{3 }= l + ^{h } ^{3n }− c
f 10
and so on.
It is easily seen that the 5th decile is the same quantity as the median. The formula for the first percentile is
h n
P_{1 }= l + − c
f 100
The formulae for the subsequent percentiles are
h 2n
P_{2 }= l + − c
f 100
h 3n
P_{3 }= l + − c
f 100
and so on. Again, it is easily seen that the 50th percentile is the same as the median, the 25th percentile is the same as the 1st quartile, the 75th percentile is the same as the 3rd quartile, the 40th percentile is the same as the 4th decile, and so on.
All these measures i.e. the median, quartiles, deciles and percentiles are collectively called quantiles. The question is, “What is the significance of this concept of partitioning? Why is it that we wish to divide our frequency distribution into two, four, ten or hundred parts?” The answer to the above questions is: In certain situations, we may be interested in describing the relative quantitative location of a particular measurement within a data set. Quantiles provide us with an easy way of achieving this. Out of these various quantiles, one of the most frequently used is percentile ranking. Let us understand this point with the help of an example.
EXAMPLE
If oil company ‘A’ reports that its yearly sales are at the 90th percentile of all companies in the industry, the implication is that 90% of all oil companies have yearly sales less than company A’s, and only 10% have yearly sales exceeding company A’s,this is demonstrated in the following figure:
Relative Frequency
It is evident from the above example that the concept of percentile ranking is quite a useful concept, but it should be kept in mind that percentile rankings are of practical value only for large data sets. It is evident from the above example that the concept of percentile ranking is quite a useful concept, but it should be kept in mind that percentile rankings are of practical value only for large data sets. The next concept that we will discuss is the graphic location of quantiles. Let us go back to the example of the EPA mileage ratings of 30 cars that was discussed in an earlier lecture.
EXAMPLE
Suppose that the Environmental Protection Agency of a developed country performs extensive tests on all new car models in order to determine their mileage rating. Suppose that the following 30 measurements are obtained by conducting such tests on a particular new car model.
When the above data was converted to a frequency distribution, we obtained:
Class Limit | Frequency |
30.0 – 32.9 | 2 |
33.0 – 35.9 | 4 |
36.0 – 38.9 | 14 |
39.0 – 41.9 | 8 |
42.0 – 44.9 | 2 |
30 |
Also, we considered the graphical representation of this distribution. The cumulative frequency polygon of this distribution came out to be as shown in the following figure:
Cumulative Frequency Polygon or OGIVE
35 30 25 20 15 10 5 0
This ogive enables us to find the median and any other quantile that we may be interested in very conveniently. And this process is known as the graphic location of quantiles. Let us begin with the graphical location of the median:
Because of the fact that the median is that value before which half of the data lies, the first step is to divide the total number of observations n by 2. In this example:
n 30
=
= 15
22
The next step is to locate this number 15 on the yaxis of the cumulative frequency polygon.
Lastly, we drop a vertical line from the cumulative frequency polygon down to the xaxis.
Cumulative Frequency Polygon or OGIVE
35 30 25 20 15 10 5
n
0
2
Now, if we read the xvalue where our perpendicular touches the xaxis, students, we find that this value is approximately the same as what we obtained from our formula.
It is evident from the above example that the cumulative frequency polygon is a very useful device to find the value of the median very quickly. In a similar way, we can locate the quartiles, deciles and percentiles. To obtain the first quartile, the horizontal line will be drawn against the value n/4, and for the third quartile, the horizontal line will be drawn against the value 3n/4.
Cumulative Frequency Polygon or OGIVE
Q1 Q3
For the deciles, the horizontal lines will be against the values n/10, 2n/10, 3n/10, and so on. And for the percentiles, the horizontal lines will be against the values n/100, 2n/100, 3n/100, and so on.
The graphic location of the quartiles as well as of a few deciles and percentiles for the dataset of the EPA mileage ratings may be taken up as an exercise: This brings us to the end of our discussion regarding quantiles which are sometimes also known as fractiles this terminology because of the fact that they divide the frequency distribution into various parts or fractions.
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