Marks 14 30 44 61 98 No. of Students 49 21 75 39 99
Calculating the Mean of a Frequency Distribution
To find the mean of a frequency distribution, we calculate the sum of the products of each value and its corresponding frequency, and then divide by the total number of observations (total frequency).
The formula used is:
$ \text{Mean} (\bar{x}) = \frac{\sum (f \times x)}{\sum f} $
Where:
- '$x$' represents the values (Marks in this problem).
- '$f$' represents the frequency of each value (No. of Students in this problem).
- '$\sum (f \times x)$' is the sum obtained by multiplying each mark by its corresponding number of students and then adding all these products together.
- '$\sum f$' is the total sum of the frequencies (total number of students).
Let's set up a table to calculate '$f \times x$' for each row and find the sums:
| Marks ($x$) | No. of Students ($f$) | Product ($f \times x$) |
|---|---|---|
| 14 | 4 | $14 \times 4 = 56$ |
| 30 | 9 | $30 \times 9 = 270$ |
| 44 | 21 | $44 \times 21 = 924$ |
| 61 | 75 | $61 \times 75 = 4575$ |
| 98 | 39 | $98 \times 39 = 3822$ |
| Total | $\sum f = 4 + 9 + 21 + 75 + 39 = 148$ | $\sum (f \times x) = 56 + 270 + 924 + 4575 + 3822 = 9647$ |
Performing the Mean Calculation
Now, we plug the total sum of products and the total frequency into the mean formula:
$ \text{Mean} = \frac{\sum (f \times x)}{\sum f} $
Substitute the calculated values:
$ \text{Mean} = \frac{9647}{148} $
Performing the division gives the mean:
$ \text{Mean} \approx 65.18 $
This detailed calculation demonstrates the step-by-step process for finding the mean of the provided distribution.
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 - If the median of a data is 61.54 less than its mode, then the median of the data exceeds its mean by _____. (Use the empirical formula to find the answer)
The mean marks of the following distribution is:
Marks Obtained No. of Students 81 15 35 4 73 3 56 16 - The geometric mean of 100 observations is 25. If each observation is multiplied by 4, what will be the new geometric mean?
