Question

Consider the following table of the ages of our first six presidents. Ages of the First 6 Presidents at Inauguration Washington 57 J. Adams 61 Jefferson 57 Madison 57 Monroe 58 J.Q. Adams 57 a. Find the mean of the data set. Round to 2 decimal places. b. Find the standard deviation (showing all the work). Do not use the statistical abilities of your calculator. You should use 2 decimal places for all non-squared number and 2 decimal places for your final answer. c. How many of the presidents’ ages fall within one standard deviation of the mean?

Answer 1

A. FINDING THE MEAN

"mean" is just the average so take the average of all the ages.  This can be done by adding up all the numbers, then dividing that by how many numbers there are.  Let's call A=average, B=all numbers added up, and C=total amount of numbers. So A=B/C where:

B=57+61+57+57+58+57=347
C=6 (because there are a total of 6 ages)

So A=347/6  ⇒  A=57.833333333333

But we need to round to 2 decimal places so we just get A=57.83

b. STANDARD DEVIATION

There are 4 steps to solving this:
1.  Work out the mean 
2. Then for each number: subtract the Mean and square the result
3. Then work out the mean of those squared differences.
4. Take the square root of that and we are done!

Step 1: We found the mean in part a. Jump to next step.
Step 2: 57-57.83= -0.83 ; 58-57.83= 0.17 ; 61-57.83= 3.17

When we square these results and round to 2 decimal places we get (-0.83)²=0.69  ; (0.17)²=0.03  ; (3.17)²=10.05

Step 3: find the mean the same way we did before.  Keep in mind 0.69 is added together 4 times.....we still need 6 numbers here

Mean=(0.69+0.69+0.69+0.69+0.03+10.05)/6
⇒ =(4(0.69)+0.03+10.05)/6
⇒ =12.84/6
⇒ = 2.14

Step 4: take the square root of the result in the previous step

√(2.14)=1.46

c. HOW MANY AGES FALL WITHIN STANDARD DEVIATION OF THE MEAN

Our mean was found to be 57.83 and our standard deviation is 1.46 so we basically just want to know which ages fall between 57.83 +/- 1.46 

57.83+1.46=59.29
57.83-1.46=56.37

So which ages fall between this range of 56.37 to 59.29? well, it's everyone who is aged 57 and 58.  When we count how many presidents this applies to we see that 5 of them fall within the deviation.  (all but J. Adams 61)