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Statistics of Grouped Data

Date: 09/12/1999 at 23:47:06
From: Tony McCabe
Subject: Statistics of Grouped Data

I am preparing to take a statistics course after many years of being 
able to avoid doing stats. I am doing some preparatory work before the 
course starts. Can you please give me answers to the following 

                      Grouped Data

     Income (*$1000)   Midpoint(x)   Number of Purchasers
     ---------------   -----------   --------------------
       20 - 29.99          25                 50
       30 - 39.99                             20
       40 - 49.99                             31
       50 - 59.99                             39
       60 - 69.99                             35
       70 - 79.99                             30
       80 - 89.99                             25
       90 - 99.99                             18

The above table is data from a survey of recent purchasers of 
superannuation plans.

1. Find the mean and standard deviation of the income of people 
purchasing superannuation plans.

     Find the mean
     Find the variance
     Find the standard deviation

2. Find the median class.

3. Choose a suitable graph and display the frequency distribution.

4. Summarize the findings.

Date: 09/13/1999 at 12:31:03
From: Doctor Mitteldorf
Subject: Re: Statistics of Grouped Data

Dear Tony,

To start: find the mean of the distribution as follows.

First, find the total number of buyers. Do this by adding up the 
column with the numbers from each income category. Second, find the 
total of all their incomes. This you can do approximately, since you 
have an estimate of the incomes of the people in the group. On each 
line, multiply the midpoint income times the number of people in the 
group. Add up the products. This should give a reasonable estimate of 
the total income. Divide total income by total buyers to give the mean 

To find the variance, you should create another column in which you 
are squaring the midpoint incomes before multiplying by the number of 
people. Add up those numbers, and divide, as before, by the total 
number of people to obtain the <mean squared income>. This is not the 
same as from the <mean income> squared - that quantity is just the 
number you calculated at first, multiplied by itself. In fact: you can 
subtract the <mean income> squared from the <mean squared income> to 
give the variance of the income distribution.

The standard deviation is the square root of the variance.

I hope that gives you a start. Dive in, try it, and report back what 
you think you understand. Explain as much as you feel comfortable 
with, and a little more. We'll try to help with a "mid-course 
correction" if you're not getting the ideas 100%.

- Doctor Mitteldorf, The Math Forum   
Associated Topics:
High School Statistics

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