The data given below shows the number of people who have saved a certain amount of money. Saving (In Rs.) Number of people 5 1 15 3 20 4 25 2 30 1 35 1 40 2 What is the median of the given data?
Rs. 20
Finding the Median of Savings Data
The question provides data showing the amount of money saved by a certain number of people. This is a frequency distribution for discrete data. To find the median of this data, we need to determine the middle value when all the savings amounts are arranged in ascending order.
Understanding the Given Data
The data is presented as pairs of Savings amount and the number of people who saved that amount (frequency).
| Saving (In Rs.) | Number of people (Frequency, \(f\)) |
|---|---|
| 5 | 1 |
| 15 | 5 |
| 20 | 3 |
| 25 | 2 |
| 30 | 3 |
| 35 | 1 |
| 40 | 2 |
Calculating the Total Number of People (N)
The total number of people is the sum of the frequencies.
Total number of people \(N = \sum f\).
\(N = 1 + 5 + 3 + 2 + 3 + 1 + 2 = 17\)
So, there are 17 observations in total.
Determining the Median Position
For a dataset with an odd number of observations \(N\), the median is the value at the \(\frac{N+1}{2}\)th position when the data is arranged in ascending order.
Median position = \(\frac{17+1}{2} = \frac{18}{2} = 9\)th term.
We need to find the savings amount corresponding to the 9th person when listed by their savings in ascending order.
Calculating Cumulative Frequency
To find which savings amount corresponds to the 9th term, we calculate the cumulative frequency. Cumulative frequency (CF) for a particular savings amount is the total number of people who saved that amount or less.
| Saving (In Rs.) | Frequency (\(f\)) | Cumulative Frequency (CF) |
|---|---|---|
| 5 | 1 | 1 (1 person saved Rs. 5 or less) |
| 15 | 5 | 1 + 5 = 6 (6 people saved Rs. 15 or less) |
| 20 | 3 | 6 + 3 = 9 (9 people saved Rs. 20 or less) |
| 25 | 2 | 9 + 2 = 11 (11 people saved Rs. 25 or less) |
| 30 | 3 | 11 + 3 = 14 (14 people saved Rs. 30 or less) |
| 35 | 1 | 14 + 1 = 15 (15 people saved Rs. 35 or less) |
| 40 | 2 | 15 + 2 = 17 (17 people saved Rs. 40 or less) |
Locating the Median Value
We are looking for the 9th term. We check the cumulative frequency column.
- The CF of 6 means the 6th person's saving is Rs. 15.
- The CF of 9 means the 9th person's saving is Rs. 20. The values from the 7th to the 9th term are Rs. 20.
- The CF of 11 means the 11th person's saving is Rs. 25.
Since the cumulative frequency becomes 9 at the saving amount of Rs. 20, the 9th term is Rs. 20.
Therefore, the median of the given data is Rs. 20.
Revision Table: Key Concepts for Median Calculation
| Concept | Description | Applicability |
|---|---|---|
| Median | The middle value of a dataset when ordered. | All types of data (quantitative) |
| Frequency (\(f\)) | Number of times a value appears. | Grouped or frequency data |
| Cumulative Frequency (CF) | Sum of frequencies up to a certain value. | Finding median/quartiles in frequency distributions |
| Median Position (Odd N) | \(\frac{N+1}{2}\)th term | Dataset with an odd number of observations |
| Median Position (Even N) | Average of \(\frac{N}{2}\)th and \((\frac{N}{2}+1)\)th terms | Dataset with an even number of observations |
Additional Information: Types of Data and Measures of Central Tendency
Types of Data
- Discrete Data: Data that can only take specific values (usually whole numbers) and cannot be divided into smaller parts meaningfully. Examples include the number of people, number of items. The savings amount here is discrete because the money values are distinct.
- Continuous Data: Data that can take any value within a given range. Examples include height, weight, time.
Measures of Central Tendency
Measures of central tendency are single values that describe the center point of a data set. The three main measures are:
- Mean: The average of all values. Calculated by summing all values and dividing by the total number of values.
- Median: The middle value in an ordered dataset. It is not affected by extreme outliers.
- Mode: The value that appears most frequently in the dataset. A dataset can have one mode (unimodal), more than one mode (multimodal), or no mode.
Choosing the appropriate measure depends on the type of data and the presence of outliers. For data with extreme values, the median is often a better measure of central tendency than the mean.
Important Questions from Elementary Statistics
What is the mode of the given data?
3, 0, 1, 0, 2, 1, 2, 0, 1, 2, 1, 1, 1, 3, 2What is the mode of the given data?
21, 22, 23, 23, 24, 21, 22, 23, 21, 23, 24, 23, 21, 23A bowler has taken 0, 3, 2, 1, 5, 3, 4, 5, 5, 2, 2, 0, 0, 1 and 2 wickets in 15 consecutive matches. What is the mode of the given data?
If the ratio of mean and median of a certain data is 4 : 5, then find the ratio of its mean and mode.
Given below is the marks obtained by 20 students in mathematics out of 30 marks.
7, 9, 12, 12, 13, 12, 14, 14, 14, 14, 15, 16, 17, 18, 18, 19, 20, 18, 20, 13. Then (2 × median — mode) of the data is equal to:
