Class 120-170 170-220 220-270 270-320 320-370 Frequency 33 13 10 31 11
Median Calculation for the Given Distribution
The median is the middle value in a dataset that separates the higher half from the lower half. For a grouped frequency distribution, we calculate the median using a specific formula after identifying the median class.
Distribution Data and Cumulative Frequency
First, let's organize the given data and calculate the cumulative frequency ($CF$) for each class interval.
| Class Interval | Frequency ($f$) | Cumulative Frequency ($CF$) |
|---|---|---|
| 120-170 | 3 | 3 |
| 170-220 | 13 | 16 |
| 220-270 | 10 | 26 |
| 270-320 | 3 | 29 |
| 320-370 | 1 | 30 |
Steps to Find the Median
-
Calculate Total Frequency ($N$):
Sum all the frequencies to find the total number of observations.
$ N = 3 + 13 + 10 + 3 + 1 = 30 $
-
Determine the Median Position:
The position of the median value is found by $\frac{N}{2}$.
$ \text{Median Position} = \frac{N}{2} = \frac{30}{2} = 15 $
This means the median is the value of the 15th observation.
-
Identify the Median Class:
Look at the cumulative frequency ($CF$) column. The median class is the class interval where the cumulative frequency first equals or exceeds the median position (15).
- The $CF$ for the 120-170 class is 3 (less than 15).
- The $CF$ for the 170-220 class is 16 (greater than or equal to 15).
Therefore, the **median class** is 170-220.
-
Apply the Median Formula:
The formula to calculate the median for grouped data is:
$ \text{Median} = L + \left( \frac{\frac{N}{2} - CF}{f} \right) \times w $
Where:
- $L$ = Lower limit of the median class
- $N$ = Total frequency
- $CF$ = Cumulative frequency of the class *preceding* the median class
- $f$ = Frequency of the median class
- $w$ = Width of the median class
-
Substitute Values and Calculate:
From our data:
- $L = 170$
- $\frac{N}{2} = 15$
- $CF = 3$ (the $CF$ of the class 120-170)
- $f = 13$ (the frequency of the median class 170-220)
- $w = 170 - 120 = 50$
Now, substitute these values into the formula:
$ \text{Median} = 170 + \left( \frac{15 - 3}{13} \right) \times 50 $
$ \text{Median} = 170 + \left( \frac{12}{13} \right) \times 50 $
$ \text{Median} = 170 + \frac{600}{13} $
$ \text{Median} \approx 170 + 46.15 $
$ \text{Median} \approx 216.15 $
The calculation shows the median value to be approximately 216.15 based on the standard formula and the provided distribution data.
Important Questions from Elementary Statistics (Notes)
Let $X_1, X_2, X_3$ be a random sample of size 3 from an absolutely continuous distribution that is symmetric about 0. For $i=1,2,3$, let $R_i$ denote the rank of $|X_i|$ among $|X_1|, |X_2|$ and $|X_3|$.
If $T^+ = \sum_{i=1, X_i>0}^3 R_i$
is the Willcoxon signed-rank statistic, then which of the following statements are true?,
- Which of the following is the first step in calculating the median of data set?
1. Average the middle two values of the data set
2. Array the data
3. Determine the relative weights of the data values in terms of importance
4. Find the average distance of the observations in the data set from the mean - The sum of deviations of the items from __________ ignoring signs is the least?
- The relationship between mean, median and mode is:
Which of the following statements are correct?
(A).When a cyclical pattern in data has a period of less than 1 year, the pattern in data is called seasonal variation
(B). When a cyclical pattern has a period more than 1 year, we refer to it as cyclical variation
(C). Seasonality is considered equivalent to forecasting
(D). Cyclical behaviour in business is also termed as business cycle
Choose the correct answer from the options given below:
