Question

Age, Deviation, Squared Deviation, Frequency, Frequency x Squared Deviation

EXERCISE #2 You fill in the 3rd column of this table. The first entry is done for you. Commas are not necessary. Frequency Table for Data Set B Age (years) Frequency Age x Frequency 18 19 20 21 1782 105 45 31 12 28 10 15 16 1995 900 651 23 24 25 26 27 28 29 30 31 35 644 2 40 3 75 416 2 16 280 2 32 1 80 155 10 6 2 70 Add the entries of the third column and put this sum in the blank. The sum of the entries in the 3td column is 8t00 Now divide this sum by 400 in order to compute the mean. Fill in the blank. The mean age of these four hundred students is 21 years (HINT: In this example the mean is a whole number) We can also find the variance and standard deviation of Data Set B by using a similar shortcut. Remember, to find the variance we use the squared deviations. To find the deviation for each age we have to subtract the mean age from each datum. In this case we will also multiply each squared deviation by its frequency

Answer 1

Following is the filled table (using the formula given in question for calculation) :-

Age X	Deviation X-mean	Squared Deviation (X-mean)^2	Frequency w	Frequency*Squared Deviation w(X-mean)^2
18	-3	9	99	891
19	-2	4	105	420
20	-1	1	45	45
21	0	0	31	0
22	1	1	12	12
23	2	4	28	112
24	3	9	10	90
25	4	16	15	240
26	5	25	16	400
27	6	36	8	288
28	7	49	10	490
29	8	64	8	512
30	9	81	6	486
31	10	100	5	500
35	14	196	2	392

Variance of Dataset B can be found as :-

$Variance = \frac{sum(w(X-mean)^2)}{400} = \frac{4878}{400} = 12.195$

So, Variance of Dataset B = 12.195

Standard Deviation of Dataset B can be found as:

$\\ Standard\ Deviation = \sqrt {Variance} = \sqrt {12.195} = 3.4921 \approx 3.5 (rounded\ to\ nearest\ tenth)$

So,  Standard Deviation of Dataset B = 3.5 (rounded to nearest tenth)