lecture notes on statistics

1. PRACTICE QUESTIONS FOR MEASURES OF CENTRAL TENDENCY
Example (the arithmetic mean p248):- The demand for a product on each of 20 days was as follows, (in
units).
3, 12, 7, 17, 3, 14, 9, 6, 11, 10, 1, 4, 19, 7, 15, 6, 9, 12, 12, 8
Calculate arithmetic mean
{Answer:- 9.25 units}
Example (grouped data p249):- From the following data, calculate arithmetic mean
Daily demand Frequency
> 0 ≤ 5 4
> 5 ≤ 10 8
> 10 ≤ 15 6
> 15 ≤ 20 2
{Answer:- 9.5 units}
Exercise 1(p 250):- For the week ended 15 November, the wages earned by the 69 operators employed in
the machine shop of Waves Ltd were as follow:
Wages Number of operative
Under Rs. 60 3
Rs. 60 and under Rs. 70 11
Rs. 70 and under Rs. 80 16
Rs. 80 and under Rs. 90 15
Rs. 90 and under Rs. 100 10
Rs. 100 and under Rs. 110 8
Rs. 110 and under Rs. 120 6
{Answer:- Rs. 84.56}
Example (weighted arithmetic mean p251):- The marks obtained by a student in Urdu, Science and
History were 60, 68 and 50 respectively. Find the appropriate average if weights of 6, 5, 4 are assigned to
these marks.
{Answer:- 60 marks}
Example (geometric mean p252):- Find the G.M of the values 4, 3, 6.
{Answer:- 4.160}
Example (geometric mean using logarithms p252):- Find the G.M for the values 3, 5, 8, 6, 7, 9, 10.
{Answer:- 6.43}
Example (geometric mean for probability distribution p253):- Find the G.M for the following
probability distribution:
X 10.20 12.30 11.50 13.00 15.70
f 1 3 4 2 3
{Answer:- 12.67}
Example (harmonic mean p253):- Find the H.M for the values 15, 20, 25, 23
{Answer:- 19.99}
Example (harmonic mean for the frequency distribution):- Find the H.M for the following frequency
distribution of weights:
Weights 20----40 41----61 62----82 83----103 104----124
f 8 7 10 6 4
{Answer:- 54.483}
Example (weighted harmonic mean):- Find the H.M of the values 20.0, 35.5, 40.0 and 37.0 with their
respective weights 7.0, 8.5, 3.0 and 6.0
{Answer:- 29.63}
Example (the mode p255):- The daily demand for stock in a ten day period is as follows:
Demand units 6 7 8
Number of days 3 6 1
{Answer:- 7}

2. Example (the formula for the mode):- Calculate the mode of the following frequency distribution
Value Frequency
At least Less than
10 25 6
25 40 19
40 55 12
55 70 7
70 85 3
{Answer:- 34.75}
Exercise 2(p 256):- An analysis of sales invoices outstanding at the end of February is as follows:
Invoice value Number of
invoicesAt least Less than
Rs. Rs.
10 25 4
25 40 8
40 55 18
55 70 9
70 85 3
{Answer:- Rs. 47.89}
Example (the median p258):- Find median of the following values:
(i) 8, 6, 9, 12, 15, 6, 3, 20, 11.
(ii) 8, 6, 7, 2, 1, 11, 3, 2, 5, 2
{Answer:- (i) 9 , (ii) 3 }
Example (median of an ungrouped frequency distribution):- Find the median of an ungrouped frequency
distribution
Value X 8 12 16 17 19
Frequency f 3 7 12 8 5
{Answer:- 16}
Example (the median of a grouped frequency distribution):- The average monthly earning of 135
employees of Honda Ltd. Have been analyzed as a grouped frequency distribution as follows:
Average monthly earnings Number of
invoicesMore than Not more than
Rs. Rs.
120 140 12
140 160 49
160 180 25
180 200 18
200 220 17
220 240 14
Calculate the median monthly earnings of employees of Honda Ltd.
{Answer:- Rs. 165.60}
Exercise 3(p 260):- The following grouped frequency distribution gives the annual wages of 200 employees
in an engineering firm.
Wages
Rs.
Number of employees
5000 and less than 5500 4
5500 and less than 6000 26
6000 and less than 6500 133
6500 and less than 7000 35
7000 and less than 7500 2
Calculate mean, median and mode of annual wages.
{Answer:- Mean = Rs. 6262.50 Median = Rs. 6263.16 Mode = Rs. 6260.98}

3. Practice question 9.1(LAHORE GARDENERS’ ASSOCIATION):- The following figures were
published by the Lahore Gardeners Association.
Earnings in 19x2 Number of
gardenersAt least Not more than
Rs. Rs.
0 500 1
500 1000 4
1000 1500 6
1500 2000 12
2000 2500 9
2500 3000 7
3000 3500 5
3500 4000 3
4000 4500 2
4500 5000 1
(a) Calculate the model earnings.
(b) Calculate the median earnings.
(c) Calculate the arithmetic mean earnings.
{Answer:- (a) Rs. 1833, (b) Rs. 2111.11, (c) Rs. 2240 }
Practice question 9.2(PAKISTAN TIMES):- (a) Set out the advantages and disadvantages of the mean,
median and mode as measure of central tendency.
{(i) THE MEAN
The mean has the following advantages;
(1) It is widely understood
(2) It uses every value in its computation
(3) It gives a unique value
It does, however, have two major disadvantages.
(1) It may give an impossible value (e.g. 2.2 children)
(2) It may be distorted by extremely high/low values.
(ii) THE MODE
Advantages of the mode are as follows
(1) It is an actual value
(2) It is not distorted by extreme values.
It has the following disadvantages.
(1) It may not be unique
(2) It ignores dispersion about the model value
(3) It does not take every value into account
(iii) THE MEDIAN
The median has the advantage that it is not distorted by extreme values but there is a disadvantange to the
median. It does not take every value into account.}

4. (b) Pakistan times to work of employees of a company were as follows
Time in Minutes No of employees
Less than 10 14
10 but less than 20 26
20 but less than 30 64
30 but less than 40 46
40 but less than 50 28
50 but less than 60 16
60 but less than 80 8
80 but less than 100 4
(i) Obtain the mean, median and mode of travel time to work
(ii) Negotiations are under consideration for payment for travelling time to work. State, giving reasons,
which measure of central tendency you would use if you are (1) the employer, and (2) a trade union
representing the employees.
{Answer: - (i) Mean = 32.67 minutes, Median = 29.84 minutes, Mode = 26.79 minutes
(ii) (1) The employer would seek to minimize paid for travelling time. He should therefore choose the mode,
perhaps arguing that the modal time is the most representative because the largest number of employees take
that long to come to work
(2) The trade union would seek to maximize the amount paid for travelling time. It should therefore choose
the mean, perhaps arguing that the mean time is the only average which takes inot account the travelling
times of all employees. }
Practice question 9.3(OUTPUT DISTRIBUTION):- Find the arithmetic mean for the following
distribution of output levels of product Q.
Output of Q kg No. of days (frequency)
350 – under 360 4
360----370 6
370----380 5
380----390 4
390----400 3
{Answer: - 373.18}
Practice question 9.4(COMPANY’S PROFIT):- The arithmetic mean of profit earned by two companies
“X” and “Y” is Rs.34 million whereas the geometric mean is Rs. 16 million. Find out the profit earned by
each company. It is known that company “X” earned more than “Y”.
{Answer: - Solution set = 64, 4}
Practice question 9.5(AVERAGE RATE OF INFLATION):-The rate of inflation in four successive years
in a country was 7 percent, 11 percent, 15 percent and 19 percent. Using the geometric mean, find the
average rate of inflation per year.
{Answer: - Average rate of inflation = 12.91%}
Practice question 9.6(SYMMETRICAL DISTRIBUTION):-
(a) In a perfectly symmetrical frequency distribution, the mean of the data is 88. Find median and mode.
(b) In a moderately skewed frequency distribution, mean = 60 and median = 55. Find mode.
{Answer: - (a) median = 88, mode = 88}