5.3.2. Discrete Series

In case of discrete series the position of median i.e. (N+1)/2th item can be located through cumulative frequency. The corresponding value at this position is the value of median.

In a discrete series, the median is calculated using the position of the data. Specifically, the position can be determined using the formula (N+1)/2, where N is the total number of observations. This formula provides the index of the median value when the data is ordered. Once we have the position, we can find the cumulative frequency, which helps locate the corresponding data value that represents the median.

Imagine you're trying to find the median height of a group of children in a classroom. First, you count how many children are there (let's say 20). To find the median, you would calculate (20+1)/2 = 10.5, which means the median height is between the 10th and 11th tallest child. This process mimics using cumulative frequency to find your specific data point.

Example 7 The frequency distribution of the number of persons and their respective incomes (in Rs) are given below. Calculate the median income. Income (in Rs): 10 20 30 40 Number of persons: 2 4 10 4 In order to calculate the median income, you may prepare the frequency distribution as given below.

In this example, we are given income levels and the corresponding number of people earning each income. To calculate the median income, we first set up a frequency distribution table. This provides a clear organizational view of how many people fall into each income category, which is essential for determining where the median lies. By preparing this table, we can easily compute cumulative frequencies, which help in finding the median value.

Consider a fundraiser where you track donations from participants: $10 from 2 people,$20 from 4 people, $30 from 10 people, and$40 from 4 people. Just like tracking donations helps to see the range of contributions, compiling this income data allows us to find out the 'median contribution', indicating what a typical donor might have given.

TABLE 5.5 Computation of Median for Discrete Series Income No. of Cumulative (in Rs) persons(f) frequency(cf) 10 2 2 20 4 6 30 10 16 40 4 20

The cumulative frequency table shows how many people have an income less than or equal to each income level. This allows us to visualize the distribution of incomes. For example, for Rs 10, there are 2 people; for Rs 20, there are 6 people (2 from Rs 10 + 4 from Rs 20), and so forth. This step is crucial because the cumulative frequency lets us find the median observation's class quickly, which helps us determine the median income.

Think of it like a race: you note how many runners have crossed certain checkpoints (income levels). The cumulative count shows you how many have completed the race at different stages, helping to identify the middle point where about half have finished.

The median is located in the (N+1)/2 = (20+1)/2 = 10.5th observation.

To find where the median value sits, we calculate its position using our earlier formula. In this scenario, we find that the median position is the 10.5th observation. This means we look between the 10th and 11th observations in our cumulative frequency table to find the value that corresponds with the median income.

Imagine splitting a group of 20 kids in a classroom to see who sits in the middle during a movie viewing. When you notice the 10th and 11th kids are the transition between the first and second halves, you'll find a representative 'middle kid' – this is like identifying our median income as we check the cumulative frequencies.

The 10.5th observation lies in the c.f. of 16. The income corresponding to this is Rs 30, so the median income is Rs 30.

From our cumulative frequency table, the 10.5th observation falls in the cumulative frequency of 16, which indicates that the median income corresponds to an income value of Rs 30. This income level represents the middle point of the dataset, meaning half of the population earns less and half earn more.

If you consider the median to be the point where half of the contributions are below and half above, then if Rs 30 is our midpoint, it's like saying that in a group of varying incomes, Rs 30 is an 'average' that balances many below and above it, just like the central position in a line of kids.

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