Question

1) The following data shows the yearly production data collected from firms in XYZ state and ABC state.

Yearly production in tons	Number of Firms in XYZ	Number of Firms in ABC
50 - 150	9	35
150 - 250	10	40
250 - 350	11	56
350 - 450	40	62
450 - 550	56	78
550 - 650	68	55
650 - 750	110	53
750 - 850	125	52
850 - 950	98	44
950 - 1050	105	42
1050 -1150	25	48
1150 -1250	20	57
1250 -1350	18	40
1350 - 1450	5	38

1) Calculate:

• the coefficient of variation for XYZ and ABC
• the interquartile range for XYZ and ABC
• the median for XYZ and ABC
• the mode for XYZ and ABC
• the variance for XYZ and ABC
• mean deviation from the mean for XYZ and ABC

Answer 1

1) Coeeficient of variation is defined as standard deviation of the data divided by the sample mean.

Let us write down general formulae for standard deviation and mean:

For XYZ,

For ABC:

b) Interquartile range is defined as Quartile3-Quartile1, where quartile3 is the point upto which the 75% of ditribution has been covered and quartile1 is the point of 25% coverage.

Quartiles are found as below:

If the number of entries is even: 2n, then the first quartile Q₁ is given by

Q₁ = median of the n smallest entries

and Q₃ = median of the n largest entries

If the number of entries is odd, 2n+1, then the first quartile Q₁ is given by

Q₁ = median of the n smallest entries

and Q₃ = median of the n largest entries

Here we have even number of entries (14), so

	XYZ	ABC
Q3	90.5	55.75
Q1	12.75	40.5
IQR	77.75	15.25

c) For even numbers, median is the average of the middle two numbers when numbers are arranged in ascending order.

Median(XYZ)=32.5

Median(ABC)=50

d) Mode refers to the most frequent outcome.

For XYZ, mode= There is no mode, since each value appears once.

For ABC, mode= 40