Chapter Questions
Find the mean and the median of: $88,72,33,29,70,54,86,91,57,61$. [C.U., B. Com. '73]
$$\begin{array}{l}\text { Find the mean, median and mode of the following numbers: } 7,4,3,5,6,3,3 \text { , }\\2,4,3,4,3,3,4,4,3,2,2,4,3,5,4,3,4,3,4,3,1,2,3 .[\text { C. } U ., \text { B.Com. '71] }\end{array}$$
Evaluate the arithmetic mean, median, and mode for the following distribution of 'number of telephone calls received per one-minute interval':$$\begin{array}{|l|l|r|r|r|r|r|r|r|r|}\hline \text { No. of Calls } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\hline \text { Frequency } & 5 & 22 & 31 & 43 & 51 & 40 & 35 & 15 & 3 \\\hline\end{array}$$
Calculate the simple and weighted average from the following and account for the difference between the two:$$\begin{array}{|l|r|r|r|}\hline \text { Price per ton }(R s) \text { . }) & 45.60 & 50.70 & 42.45 \\\hline \text { Tons Purchased } & 135 & 40 & 25 \\\hline\end{array}$$
The numbers $3.2,5.8,7.9$ and $4.5$ have frequencies $x,(x+2),(x-3)$ and $(x+6)$ respectively. If the arithmetic mean is $4.876$, find the value of $x$.
$$\text { Calculate the arthmetic mean from the following data: }$$$$\begin{array}{|llccccccc|}\hline \text { (i) } & \text { Class Interval } & 50-59 & 60-69 & 70-79 & 80-89 & 90-99 & 100-109 & 110-119 \\& \text { Frequency } & 14 & 38 & 44 & 54 & 45 & 30 & 25 \\\hline \text { (ii) } & \text { Height in Inches } & 57.5 & -60.0 & -62.5 & -65.0 & -67.5 & -70.0 & -72.5- \\& \text { Number of Men } & 6 & 26 & 190 & 281 & 412 & 127 & 38 \\\hline \text { (iii) } & \text { Weight in lbs. } & 137.5-147.5 & 147.5-157.5 & 157.5-167.5 & \\& \multicolumn{1}{l} {\text { Number of Men }} & \multicolumn{2}{c} {5} & \multicolumn{2}{c|} {4} & \\& 167.5-177.5 & 177.5-187.5 & 187.5-197.5 & \multicolumn{2}{c|} {197.5-217.5} & 217.5-247.5 \\\multicolumn{10}{|c} {5} & 7 & 5 & 3 & 1 \\\hline \text { (iv) } & X \quad 20-30 & 30-50 & 50-100 & 100-200 & 200-350 & 350-550 \\& \text { Freq. } \quad 2 & 9 & 11 & 52 & 10 & 3 \\\hline\end{array}$$
From the data in the following table calculate the average marks of the M.Com. examinees in Statistics at a class test:\begin{tabular}{|l|c|c|c|c|c|c|c|}\hline Marks & $30-39$ & $40-49$ & $50-59$ & $60-69$ & $70-79$ & $80-89$ & $90-99$ \\\hline No. of & & & & & & & \\Examinees & 2 & 3 & 11 & 20 & 32 & 25 & 7 \\\hline\end{tabular}
The following table gives the rise in price of 300 commodities between two dates. Calculate the mean rise in price:\begin{tabular}{|l|l|l|l|l|l|l|l|c|}\hline$\%$ Increase & $0-$ & $5-$ & $10-$ & $15-$ & $25-$ & $35-$ & $45-$ & $60-80$ \\\hline Frequency & 12 & 30 & 51 & 84 & 66 & 35 & 15 & 7 \\\hline\end{tabular}
The following are the monthly salaries (in $\mathrm{Rs}$ ) of 30 employees in a firm:\begin{tabular}{|llllllllll|}\hline 140 & 139 & 126 & 114 & 100 & 88 & 62 & 77 & 99 & 103 \\108 & 129 & 144 & 148 & 134 & 63 & 69 & 148 & 132 & 118 \\142 & 116 & 123 & 104 & 95 & 80 & 85 & 106 & 123 & 133 \\\hline\end{tabular} The firm gave bonus of $\mathrm{Rs} 10,15,20,25,30,35$ for individuals in the respective salary groups: 'exceeding Rs 60 but not exceeding Rs $75^{\prime}$; 'exceeding Rs 75 but not exceeding $\mathrm{Rs} 90^{\prime} ;$ and so on upto 'exceeding $\mathrm{Rs} 135$ but not exceeding Rs 150 '. Find the average bonus paid per worker.
For the variable $x$, taking the values $0,1,2, \ldots, k$, the cumulative frequencies of more-than type are $F_{0}, F_{1}, F_{2}, \ldots, F_{k}$. Show that $\bar{x}=\sum_{i=1}^{k} F_{i} \mid n$, where $n$ is the total frequency.
(a) The arthmetic mean calculated from the following frequency distribution is known to be $67.45$ inches. Find the value of $f_{3}$. \begin{tabular}{|l|c|c|c|c|c|}\hline Height (inches) & $60-62$ & $63-65$ & $66-68$ & $69-71$ & $72-74$ \\\hline Frequency & 15 & 54 & $f_{3}$ & 81 & 24 \\\hline\end{tabular} [I.C.W.A., July '71](b) The expenditure of 1000 families is given below:\begin{tabular}{|l|c|c|c|c|c|}\hline Expenditure $(R s):$ & $40-59$ & $60-79$ & $80-99$ & $100-119$ & $120-139$ \\\hline No. of Families: & 50 & $?$ & 500 & $?$ & 50 \\\hline\end{tabular} The median and mean for the distribution are both Rs $87.50$ P. Calculate the icsinc frecuencies
Find out the missing frequencies of the following data, given that A.M. is $67.45$ inches. \begin{tabular}{|l|c|c|c|c|c|c|}\hline Height (inches) & $60-62$ & $63-65$ & $66-68$ & $69-71$ & $72-74$ & Total \\\hline No. of Students & 5 & 18 & $f_{3}$ & $f_{4}$ & 8 & 100 \\\hline\end{tabular}
Out of the total population in a certain town in South Africa, $60 \%$ belonged to the Black Race and the rest belonged to the White Race. It was estimated that their mean incomes were respectively 2,000 and 5,000 pounds. Find the average income of the entire town 681
(i) A factory has five sections employing $105,184,130,93$ and 125 workers. The mean earnings in a certain week per worker are Rs $13.80$, Rs $15.00$, Rs $15.20$, Rs $18.20$ and Rs $14.20$ for the 5 sections. Determine the mean earning per worker of the whole factory. [Dip. Management, '70](ii) In a survey of locality the following figures regarding the income of the people in different occupations were received. Find out the average per capita income:$$\begin{array}{|l|c|c|}\hline \text { Occupation } & \begin{array}{c}\text { Average Income } \\(\text { in } R s)\end{array} & \text { \mathrm{Number of People } } \\\hline \text { Business } & 500 & 700 \\\text { Labour } & 300 & 300 \\\text { Craftmanship } & 200 & 200 \\\text { Other } & 400 & 100 \\\hline\end{array}$$
The following shows some data collected for three regions of a country:\begin{tabular}{|c|c|c|c|}\hline Region & No. of Inhabitants (million) & Percentage of Literates & Average Anmual Income per Person $(R s)$ \\\hline A & 10 & 52 & 850 \\B & 5 & 68 & 620 \\C & 18 & 39 & 730 \\\hline\end{tabular} Obtain the over-all figures for the three regions taken together.
The population of India in 1951 and in 1961 were 361 and 439 million respectively. (i) What was the average percentage increase per year during the period? (ii) If the average rate of increase from 1961 to 1971 remains the same. what would be the population in $1971 ?$
A man gets three successive annual increments in salary of $20 \%, 30 \%$ and $25 \%$, each percentage being reckoned on his salary at the end of the previous year. How much better or worse off would he have been if he had been given 3 annual increments of $25 \%$ each, reckoned in the same way?
A machine is assumed to depreciate $40 \%$ in value in the first year, $25 \%$ in the second year and $10 \%$ per annum for the next 3 years, each percentage being calculated on the diminishing value. What is the average percentage depreciation, reckoned on the diminishing value, for the 5 years?
The G.M. of 4 observations is 47 , and the G.M. of 6 others is 40 . Find the G.M. of all the 10 observations.
The geometric mean of six numbers is 75 . If the geometric mean of four of them is 67 , what is the geometric mean of the other two?
You fly to a place $X$ in a Boeing at a speed of 500 miles per hour and come back from $X$, following the same route, at a speed of $160 \mathrm{~m} . \mathrm{p} \cdot \mathrm{h}$. What is your average speed for the to-and-fro journey?
An aeroplane flies around a square the sides of which measure $100 \mathrm{kms}$. each. The aeroplane covers at a speed of $100 \mathrm{kms}$. per hour the first side, at $200 \mathrm{kms}$ per hour the second side, at $300 \mathrm{Kms}$. per hour the third side, and at $400 \mathrm{kms}$ per hour the fourth side. Use the correct mean to find the average speed round the square.
If two grades of oranges sell at 10 for Re. 1 and 20 for Re. 1 respectively, calculate the average price per orange. stating your assumptions explicitly.
(a) The weights (in lb.) of 8 persons are: $138,143,141,139,152,148,160$ and 267 . Find the average weight using a suitable form of average. Give reasons for your choice.(b) What is the suitable form of average in each of the following cases?(i) The length of a rod is measured by a tape 10 times. You are to estimate the length of the rod by averaging these 10 determinations. (ii) A person purchases 5 rupees worth of eggs from 10 different markets. You are to find the average number of eggs per rupee for all the markets taken together.(iii) You are given the population of India for the censuses of 1961 and1971. You are to find the population of India at the middle of the period by averaging these population figures, assuming a constant rate of increase of population.
Find the mean and the median for the following data, and comment on the shape of the distribution:\begin{tabular}{|l|c|c|c|c|c|c|c|}\hline Weight in kg & $36-40$ & $41-45$ & $46-50$ & $51-55$ & $56-60$ & $61-65$ & $66-70$ \\\hline No. of Persons & 14 & 26 & 40 & 53 & 50 & 37 & 25 \\\hline\end{tabular}
The G.M., H.M. and A.M. of three observations are $3.63,3.27$ and 4 respectively. Find the observations. $\quad[$ C.U., M.Com. '75]
$$\begin{array}{l}\text { Using a suitable formula calculate the median value from the following data: }\\\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|}\hline \text { Midvalue } & 115 & 125 & 135 & 145 & 155 & 165 & 175 & 185 & 195 & \text { Total } \\\hline \text { Frequency } & 6 & 25 & 48 & 72 & 116 & 60 & 38 & 22 & 3 & 390 \\\hline\end{array}\end{array}$$
In a group of 1000 wage earners the monthly wages of $4 \%$ are below $\operatorname{Rs} 60$ and those of $15 \%$ are under $\mathrm{Rs} 62.50 .15 \%$ earned $\mathrm{Rs} 95$ and over, and $5 \%$ gotRs 100 and over. Find the median wage
The table below gives the frequency distribution of weights of 80 applies:\begin{tabular}{|l|c|c|c|c|c|c|c|c|}\hline Weight $(\mathrm{gms})$ & $110-119$ & $120-129$ & $130-139$ & $140-149$ & $150-159$ & $160-169$ & $170-179$ & $180-189$ \\\hline Frequency & 5 & 7 & 12 & 20 & 16 & 10 & 7 & 3 \\\hline\end{tabular} Draw the cumulative frequency diagram and hence determine the median weight of an apple.
Draw the less than Ogive and estimate the value of median on the basis of the data given below:\begin{tabular}{|l|r|r|r|r|r|r|r|r|}\hline \mathrm{Mid-Point } & 18 & 25 & 32 & 39 & 46 & 53 & 60 & \\\hline Frequency & 10 & 15 & 32 & 42 & 26 & 12 & 9 & $N=146$ \\\hline\end{tabular}
An incomplete frequency distribution is given below:\begin{tabular}{|l|c|c|c|c|c|c|c|}\hline Height (inches) & $5.1-6.0$ & $6.1-7.0$ & $7.1-8.0$ & $8.1-9.0$ & $9.1-10.0$ & $10.1-11.0$ & $11.1-12.0$ \\\hline No. of Plants & 3 & 8 & 27 & $?$ & 17 & 11 & 9 \\\hline\end{tabular} It is known that the median height of the plant is $8.53$ inches. Calculate the $\begin{array}{ll}\text { missing frequency. } & \text { [I.C.W.A., Jan. '72] }\end{array}$
In each of the following cases, explain whether the description applies to the mean, median or both:(i) can be calculated from a frequency distribution with open end intervals.(ii) the values of all items are taken into consideration in the calculation,(iii) the values of extreme items do not influence the average,(iv) in a distribution with a single peak and moderate skewness to the right, it is closer to the concentration of the distribution. $\quad[$
Calculate the value of the mode by the usual formula (after grouping if necessary):\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\hline$x$ & $10-20$ & $20-30$ & $30-40$ & $40-50$ & $50-60$ & $60-70$ & $70-80$ & $80-90$ & $90-100$ & $100-110$ \\\hline$f$ & 4 & 6 & 5 & 10 & 20 & 22 & 24 & 6 & 2 & 1 \\\hline\end{tabular}
From the following distribution of weekly earnings, calculate (i) the most usual wage, and (ii) the percentage earning more than Rs $31.50$. \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}\hline Weekly Earnings $(R s):$ & $25-$ & $26-$ & $27-$ & $28-$ & $29-$ & $30-$ & $31-$ & $32-$ & \\\hline No. of Persons: & 25 & 70 & 210 & 275 & 430 & 550 & 340 & 130 & \\\hline & & & & & $33-$ & $34-$ & $35-$ & 36 & Total \\\hline & & & & & 90 & 55 & 25 & & 2200 \\\hline\end{tabular}
$$\begin{array}{l}\text { Find the mean and mode for the following: }\\\begin{array}{|l|c|c|c|c|c|c|}\hline \text { Year under } & 10 & 20 & 30 & 40 & 50 & 60 \\\hline \text { Number of Persons } & 15 & 32 & 51 & 78 & 97 & 109 \\\hline\end{array}\end{array}$$
From the following cumulative frequency distribution of marks obtained by 22 students, calculate (a) Arithmetic mean, (b) Median, and (c) Mode. \begin{tabular}{|c|c|}\hline Marks & No. of Students \\\hline Below 10 & 3 \\Below 20 & 8 \\Below 30 & 17 \\Below 40 & 20 \\Below 50 & 22 \\\hline\end{tabular}
The table below gives the numbers $(f)$ of candidates obtaining marks $x$ or higher in a certain examination (all marks are given in whole numbers):\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}\hline$x$ & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 & 90 & 100 \\\hline$f$ & 140 & 133 & 118 & 100 & 75 & 45 & 25 & 9 & 2 & 0 \\\hline\end{tabular}
Calculate the values of (i) mean, (ii) median, and (iii) the two quartiles:Income \begin{tabular}{|l|c|c|c|c|c|c|c|c|c|}\hline$(R s$ 'ooo): Under & 1 & $1-2$ & $2-3$ & $3-5$ & $5-10$ & $10-25$ & $25-50$ & $50-100$ & $100-1000$ \\\hline No. of Persons: & 13 & 90 & 81 & 117 & 66 & 27 & 6 & 2 & 2 \\\hline\end{tabular}
In a moderately asymmetrical distribution, the mean and the median are respectively $25.6$ and $26.1$ inches. What is the mode of the distribution?
In a moderately skewed distribution, A. Mean $=24.6$, and the mode $=26.1$. Find the value of the Median and explain the reason for the method employed.
$$\text { Compute the median and the upper quartile of the following: }$$$$\begin{array}{|lrrrrrrrr|}\hline \multicolumn{1}{|l|} {\text { Intelligence Quotient }(I Q):} \\\multicolumn{1}{|l|} {55-64} & 65-74 & 75-84 & 85-94 & 95-104 & 105-114 & 115-124 & 125-134 & 135-144 \\\hline \multicolumn{1}{|l|} {\text { No. of Students: }} \\2 & 20 & 79 & 184 & 302 & 207 & 82 & 24 & 4 \\\hline\end{array}$$
The weekly wages earned by the hundred workers of a factory are set out in the following table:$$\begin{array}{|lccccc|}\hline \text { Weekly Wages }(R s): & \multicolumn{2}{c} {12.5-17.5} & \multicolumn{1}{c} {17.5-22.5} & 22.5-27.5 & \multicolumn{2}{c|} {27.5-32.5} \\\hline \text { No. of Workers: } & \multicolumn{2}{c} {12} & \multicolumn{2}{c} {16} & 25 & \multicolumn{2}{c|} {14} \\& 32.5-37.5 & 37.5-42.5 & 42.5-47.5 & 47.5-52.5 & 52.5-57.5 \\& 13 & 10 & 6 & 3 & 1 \\\hline\end{array}$$
The following table shows the age distribution of heads of families in a certain country during the year 1957 . Find the median, the third quartile and the second decile of the distribution. Check your results by the graphical method:
For an income distribution of a group of men 20 per cent of men have income below Rs 30,35 per cent below Rs 70,60 per cent below Rs 150 and 80 per cent below Rs 250 . The first and third quartiles are Rs 50 and Rs 170 . Put the above information in a cumulative frequency distribution and find the median.
"For a group of 5000 workers the weekly wages vary from Rs 20 to $\mathrm{Rs} 80$. The wages of 4 per cent of the workers are under Rs 25 and those of 10 per cent are under Rs $30 ; 15$ per cent of the workers earn Rs 60 and over, and 5 per cent of them get Rs 70 and over. The quartile wages are Rs 40 and $\mathrm{Rs} 54$, and the sixth decile is Rs $50^{\prime \prime}$. Put the above information in the form of a frequency distribution and find the mean wage therefrom.
For a certain group of 'Saree' weavers of Varanasi, the median and quartile of earnings per week are Rs $44.30$, Rs $43.00$ and Rs $45.90$ respectively. Ten per cent of the group eam under Rs 42 per week and $13 \%$ earn Rs 47 and over, and $6 \%$ Rs 48 and over. The range of earnings per week is Rs 40 -Rs 50 . Put the data into a frequency distribution. $\quad
Comment on the following statements:(a) "The median of a distribution is $N / 2$, the lower quartile is $N / 4$ and the upper quartile is $3 N / 4 "$, (Here $N$ denotes the total frequency). [I.C.W.A. Dec. '74](b) "If $Q_{1}, Q_{2}, Q_{3}$ be respectively the lower quartile, median and the upper quartile of a distribution, then $Q_{2}-Q_{1}=Q_{3}-Q_{2} .$ [I.C.W.A.June '75]
Given below is the frequency distribution of carbon content (present) in 150 determinations of a certain mixed powder.$$\begin{array}{l}\begin{array}{|l|c|c|c|c|c|c|}\hline \text { Per cent Carbon: } & 4.0-4.1 & 4.2-4.3 & 4.4-4.5 & 4.6-4.7 & 4.8-4.9 & 5.0-5.1 \\\hline \text { Frequency: } & 1 & 2 & 7 & 20 & 25 & 30 \\\hline & & & & 5.2-5.3 & 5.4-5.5 & 5.6-5.7 \\\hline & & & & 10 & 25 & 30 \\\hline\end{array}\\\text { Compute the arithmetic mean and median. }\end{array}$$
(i) The mean, median and mode are located at the same point in a frequency distribution.(ii) Extreme values in the series affect the utility of the mean but not the median or mode. $\quad$ [D.S.W., '78]
Compute the arithmetic mean, median and mode of the following distribution and explain their relationships:$$\begin{array}{|c|c|c|c|c|c|c|}\hline \begin{array}{l}\text { Monthly } \\\text { income }(R s)\end{array} & 0-75 & 75-150 & 150-225 & 225-300 & 300-375 & 375-450 \\\hline \text { Frequency } & 15 & 200 & 250 & 225 & 10 & 5 \\\hline\end{array}$$
(a) If $z_{1}=x_{1}+y_{1}, z_{2}=x_{2}+y_{2}, \ldots z_{n}=x_{n}+y_{n}$, then prove that $\bar{z}=\bar{x}+\bar{y}$,where the symbols have their usual meaning.(b) Prove that the logarithm of geometric mean of observations is the arithmetic mean of logarithms of the observations.
$$\begin{array}{l}\text { The following are the population figures (in thousands) of } 10 \text { cities. Find the }\\\text { median: } 2488,1490,777,733,522,672,591,407,387 \text { and 391. [D.M., '78] }\end{array}$$