Math Problem Statement

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.

Solution

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

AgePension (Rs.)
74300
62200
84400
72300
61200
83400
72300
81400
64200
71300
63200
61200
60200
67250
74300
64200
79350
73300
75350
76350
69250
68250
78350
66250
67250

Step 2: Calculate the Monthly Average Pension

To find the average pension:

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

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\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:

σ=1Ni=1N(xiμ)2\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2}

Where:

  • NN is the number of persons (25 in this case),
  • xix_i is the pension amount for each person,
  • μ\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.

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Math Problem Analysis

Mathematical Concepts

Statistics
Average
Standard Deviation

Formulas

Average calculation
Standard deviation formula

Theorems

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Suitable Grade Level

Advanced High School