A charitable organisation decided to give old age pensions to people over sixty years 
of
age. The scale of pensions were fixed as follow:
Age group 60—65 Rs. 200 per month
” 65—70 Rs. 250 per month
” 70—75 Rs. 300 per month
” 75—80 Rs. 350 per month
” 80—85 Rs. 400 per month
The ages of 25 persons who secured the pensions right are as given below:
74, 62, 84, 72, 61, 83, 72, 81, 64, 71, 63, 61, 60,
67, 74, 64, 79, 73, 75, 76, 69, 68, 78, 66, 67
Calculate monthly average pensions payable per person and the standard deviation.

To solve this problem, we need to first determine the pension amount for each of the 25 persons based on their ages, then calculate the monthly average pension per person, and finally, compute the standard deviation of the pensions.

### Step 1: Assign Pension Amounts Based on Age
We'll start by assigning the pension amounts to each person according to their age group:

- **Age 60—65**: Rs. 200 per month
- **Age 65—70**: Rs. 250 per month
- **Age 70—75**: Rs. 300 per month
- **Age 75—80**: Rs. 350 per month
- **Age 80—85**: Rs. 400 per month

Given ages: 74, 62, 84, 72, 61, 83, 72, 81, 64, 71, 63, 61, 60, 67, 74, 64, 79, 73, 75, 76, 69, 68, 78, 66, 67

| Age | Pension (Rs.) |
|-----|---------------|
| 74  | 300           |
| 62  | 200           |
| 84  | 400           |
| 72  | 300           |
| 61  | 200           |
| 83  | 400           |
| 72  | 300           |
| 81  | 400           |
| 64  | 200           |
| 71  | 300           |
| 63  | 200           |
| 61  | 200           |
| 60  | 200           |
| 67  | 250           |
| 74  | 300           |
| 64  | 200           |
| 79  | 350           |
| 73  | 300           |
| 75  | 350           |
| 76  | 350           |
| 69  | 250           |
| 68  | 250           |
| 78  | 350           |
| 66  | 250           |
| 67  | 250           |

### Step 2: Calculate the Monthly Average Pension
To find the average pension:

\[
\text{Average Pension} = \frac{\text{Sum of all pensions}}{\text{Number of persons}}
\]

\[
\text{Sum of all pensions} = 300 + 200 + 400 + 300 + 200 + 400 + 300 + 400 + 200 + 300 + 200 + 200 + 200 + 250 + 300 + 200 + 350 + 300 + 350 + 350 + 250 + 250 + 350 + 250 + 250
\]

Let's calculate this sum and then find the average.

### Step 3: Calculate the Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion of a set of values. We'll use the formula for standard deviation of a sample:

\[
\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2}
\]

Where:
- \(N\) is the number of persons (25 in this case),
- \(x_i\) is the pension amount for each person,
- \(\mu\) is the average pension.

Let's compute the sum, average, and standard deviation.

I'll perform these calculations now.### Results:
1. **Total sum of pensions**: Rs. 7050
2. **Monthly average pension per person**: Rs. 282.00
3. **Standard deviation of the pensions**: Rs. 67.65

These results indicate that the average pension distributed to each person is Rs. 282, with a variation (standard deviation) of Rs. 67.65 around this average.

Would you like any further details or explanations?

Here are five related questions:
1. How does the standard deviation affect the distribution of pensions?
2. What would be the impact on the average pension if another age group was added?
3. How would the results change if the sample size were larger?
4. Can we calculate the median pension, and how does it compare to the mean?
5. What would the results look like if the pension amounts were different?

**Tip:** Understanding the standard deviation helps in assessing the spread or variability in any data set, which is crucial for making informed decisions based on that data.

Learn how to calculate the monthly average pensions and standard deviation for a charitable organization's pension distribution based on age groups. This problem involves statistics, including average calculation and standard deviation formula, applied to pension amounts allocated to individuals aged over 60 years. Discover the average pension per person and the variation around this average. Suitable for advanced high school students and anyone interested in statistical analysis.

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