Question

Q1.The following table shows the age distribution of live births in Albany County, New York, for 2000.​

Mother’s Age

Number of Deaths due to Accidents

10-14

7

15-19

258

20-24

585

25-29

841

30-34

981

35-39

526

40-44

99

45-49

4

(a) For the data Calculate Mean, median and Mode.

(b) Which average is most appropriate and why?

(c) Comment on the Shape of the distribution from (a).​

Q2. Calculate the arithmetic mean taking A=8.95 by (i) Short-cut Method and (ii) Coding Method for the following data.

Death Rate

3.5-4.4

4.5-5.4

5.5-6.4

6.5-7.4

7.5-8.4

8.5-9.4

9.5-10.4

10.5-11.4

11.5-12.4

12.5-13.4

13.5-14.4

F

1

4

5

13

12

19

13

10

6

4

1

Answer 1

1.)

So, from the table:

$Mean=\dfrac{\Sigma f_i\cdot x_i}{\Sigma f_i}=\dfrac{95247}{3301}=28.85$

Next, $N/2 = 3301/2 = 1650.5$

So, median class is 24.5-29.5

Then, $l=24.5, h=5,c.f.=850,and\ f=841$

$Median=l+(\dfrac{\frac N2-cf}{f})\cdot h=24.5+(\dfrac{1650.5-850}{841})\times 5=29.26$

Next, modal class is 29.5-34.5

$f_1=981,f_0=841,f_2=526,and\ l=29.5$

So, $mode=l+(\dfrac{f_1-f_0}{2f_1-f_0-f_2})\cdot h=29.5+(\dfrac{981-841}{2(981)-841-526})\times 5=30.67$

(b) Median is the most appropriate average as it is a skewed distribution. For a skewed distribution, the mean tends to be biased due to extreme values.

(c)  As here mean < median < mode, so it is a negatively skewed distribution.

(2.)

(i) Short-cut method:

Arithmetic mean using shortcut method,

Now, $\bar x=A+\dfrac{\sum F d}{\sum F}$

$\Rightarrow \bar x =8.95+\frac{(-2)}{88}=8.927$

Therefore arithmetic mean taking A=8.95 by Short-cut Method is 8.927.

(ii) Coding Method :

Now, $\bar x=A+\dfrac{\sum fU}{\sum f}$

$\Rightarrow \bar x=8.95+\dfrac{(-2.222)}{88}\\\Rightarrow \bar x=8.9247$

Using coding method, arithmetic mean of the data is 8.9247.