Question 11 Mark
The number of trips made by $24$ taxi drivers in a day are shown in the following data. Find the modal number of trips.
AnswerThe maximum frequency is $7$ which is the frequency for the observations $2$ and $5$ .
Here, we say that this distribution has two modes $M_{o}=2$ and $M_{o}=5$.
Thus, the modes for the number of trips by the taxi driver are $2$ and $5$ .
Note : Such a distribution is called as a bimodal distribution. Similarly there can be distributions with more than two modes.
View full question & answer→Question 21 Mark
$TV$ sets assembled by a $TV$ manufacturing company in a month are tested. The following table shows the numbers of defects per $TV$ set. Find the mode for the number of defects.
AnswerAn inspection of frequencies shows that the observation $0$ has the maximum frequency $45$.
Hence $M_{o}=0$
Thus the mode for the number of defects in $TV$ sets is $0$.
Note : According to the definition, the value of mode in this illustration is $0$. But it cannot be taken as a measure of central tendency for the data as the value of mode is in the beginning of the data.
Mean or Median should be chosen as the measures of central tendency for such data or the value of mode should be found using empirical formula based on the values of mean and median which will be discussed in the later part of this chapter.
View full question & answer→Question 31 Mark
The numbers of books purchased by each of the $15$ persons from a book store are as follows.
$1, 0, 2, 2, 3, 4, 2, 7, 2, 2,5, 4,2,1, 2.$
Find the modal value for the number of books purchased.
AnswerWe can see that the value of $2$ is repeated $7$ times which is more than the number of repetitions of any value of the other observations. Hence mode $M_{o}=2$.
Thus, the mode of the number of books purchased is $2$ .
View full question & answer→Question 41 Mark
Find the arithmetic mean and geometric mean of two numbers $9$ and $16$ and verify that $\bar{x}>G$
AnswerHere $x_{1}=9, x_{2}=16$ and $n=2$
Arithemetic mean $\bar{x}=\frac{\sum x}{n}=\frac{9+16}{2}=\frac{25}{2}=12.5$
Geometric mean $G=\sqrt{x_{1} \times x_{2}}=\sqrt{9 \times 16}=\sqrt{144}=12$
Since $\bar{x}=12.5$ and $G=12$ we can see that $\bar{x}>G$
View full question & answer→Question 51 Mark
The geometric mean of two numbers is $2$. If one number is $4$ times the other number, find the numbers.
AnswerSuppose the smaller number is $x_{1}=a$.
Then the larger number which is 4 times this number will be $x_{2}=4 a$ and $G=2$.
$ \begin{aligned} \quad G &=\sqrt{x_{1} \times x_{2}} \\ \therefore \quad 2 &=\sqrt{a \times 4 a} \\ \therefore \quad 2 &=\sqrt{4 a^{2}} \\ \therefore \quad 2 &=2 a \\ \therefore \quad a &=1 \end{aligned} $
Thus, we get the first number $x_{1}=a=1$ and the second number $x_{2}=4 a=4$.
View full question & answer→Question 61 Mark
The following data show the number of scooters repaired daily at a garage. Find the mean number of scooters repaired per day.
$7, 13, 4, 8, 6, 9, 10, 4$
AnswerHere $n=8$
Mean$\begin{aligned} \bar{x} &=\frac{\Sigma_{x}}{n}=\frac{x_{1}+x_{2}+\ldots+x_{8}}{8} \\ &=\frac{7-13-4-8-6-9-10-4}{8} \\ &=\frac{61}{8} \\ &=7.625 \\ &=7.63 \end{aligned}$
Thus, the mean number of scooters repaired per day at the garage is $7.63$ scooters.
View full question & answer→Question 71 Mark
Mean of variable x in $9$. What is the mean of the variable $y = x + 4 ?$
Answer$\overline{ x }=17, y = x -4$
$\therefore \overline{ y }=\overline{ x }-4=17-4=13$
View full question & answer→Question 81 Mark
Median of $10$ observations is $55$. If the value of the largest observation increases from $100$ to $110$, find the new median.
Answer$n = 10, M = 55$. The largest observation increases from $100$ to $110$.
The value of median is free from the effect of extreme values. Therefore.
the new value of median remains same as $55.$
View full question & answer→Question 91 Mark
State the name of the statistician who gave the empirical formula between mean, median and mode.
AnswerThe noted statistician Karl Pearson gave the empirical formula between mean, median and mode.
View full question & answer→Question 101 Mark
AnswerThe value that occurs maximum number of times in the given data is called mode. It is denoted by $M_0$.
View full question & answer→Question 111 Mark
State the condition under which geometric mean cannot be found.
AnswerWhen the value of any one variable is either zero or negative, then geometric mean cannot be found.
View full question & answer→Question 121 Mark
State the empirical relation between mean, median and mode.
AnswerThe empirical relation between mean, median and mode is $M_0=3 M-2$.
Where, $\mathrm{M}_0=$ Mode; $\mathrm{M}=$ Median; = Mean.
View full question & answer→Question 131 Mark
Name any two positional averages.
AnswerTwo positional averages are :
$(1)$ Median and
$(2)$ Quartiles.
View full question & answer→Question 141 Mark
If observations have varying importance, which average should be used ?
AnswerIf observations have varying importance, weighted average should be used.
View full question & answer→Question 151 Mark
The median of daily demand of a vendor is $15$. If he sells each item for $Rs. 10$. find the median of his revenue ?
Answer$M= 15,$
The value of each item $= Rs. 10$
Therefore, median income of item $= Rs. (15 \times 10 ) = Rs. 150$
Hence, median income $= Rs. 150$
View full question & answer→Question 161 Mark
If $Q_3 = 25.75$ for a variable, then find $P_{75}.$
Answer$Q_3=25.75 \therefore P_{75}=Q_3=25.75$
View full question & answer→Question 171 Mark
Which average can be obtained if the continuous frequency distribution has open ended classes ?
AnswerIf the continuous frequency distribute has open ended classes median can be obtained as an average.
View full question & answer→Question 181 Mark
Find first quartile for variable with observations $15, 4, 7, 20, 2, 7, 13.$
AnswerArranging the observations in ascending order $: 2, 4, 7, 7, 13, 15, 20$
Here. $N = 7.$
First quartile $Q _1=$ Value of $\left(\frac{n+1}{4}\right)$ th observation $=$ Value of $\left(\frac{7+1}{4}\right)=2$ nd observation
$Q_1=4$
View full question & answer→Question 191 Mark
Arithmetic mean of two numbers is $5$. If one number is $6$, find the other number.
AnswerSuppose, another number is $x_2$
$\bar{x}=\frac{6+x 2}{2}$
$\therefore 5=\frac{6+x 2}{2} $
$\therefore=10=6+x_2$
$\therefore x _2=10-6=4$
Hence, another number $= 4.$
View full question & answer→Question 201 Mark
Find the modal value of the variable having the following frequency distribution :
| $X$ |
$5$ |
$10$ |
$15$ |
$20$ |
$25$ |
| $f$ |
$12$ |
$48$ |
$23$ |
$10$ |
$2$ |
AnswerMode $\mathrm{M}_0=$ Observation having a maximum frequency $48$
$\therefore \mathrm{M}_0=10$
View full question & answer→Question 211 Mark
State any one advantage of mean.
AnswerThe value of mean is based on all the observations of the data.
View full question & answer→Question 221 Mark
The$ IQ$ levels of students in a class are given below. Find the modal value of $IQ$ level of students.
$146, 134, 143, 144, 138, 145, 153, 138, 138, 146, 140, 135$
AnswerIn the given data, observation $138$ repeats thrice which is the highest than the repetition of any other observation of data.
Hence, the mode of intelligence quotent of student $M_0$ $= 138$
View full question & answer→Question 231 Mark
The weekly growths (in $cm$) in saplings grown in a nursery are :
$1.0, 3.2, 1.4, 1.9, 2.4, 1.6, 1.4, 2.1, 1.3, 1.5$
Find the mean growth.
AnswerHere, $n=10 ; x=$ weekly growth (in $\mathrm{cm}$ ) of plant
Mean of growth of plant :
$\overline{\mathrm{X}} =\frac{\Sigma x}{n}$
$ =\frac{1.0+3.2+1.4+1.9+2.4+1.6+1.4+2.1+1.3+1.5}{10}$
$ =\frac{17.8}{10}$
$ =1.78 \mathrm{~cm}$
View full question & answer→Question 241 Mark
Comment on the mode for the following data showing the time taken (in seconds) for competitors in a running race:
$25.2, 26.5, 28.6, 32.1, 29.0, 29.3, 31.3, 27.8$
AnswerMode cannot be found using its definition but it can be found using empirical formula.
View full question & answer→Question 251 Mark
The $IQ$ levels of students in a class are given below. Find the modal value of $19$ level of students.
$146, 134, 143, 144, 138, 145, 153, 138, 138, 146, 140, 135$
View full question & answer→Question 261 Mark
The weekly growths (in $cm$) in saplings grown in a nursery are:
$1.0, 3.2, 1.4, 1.9, 2.4, 1.6, 1.4, 2.1, 1.3, 1.5$
Find the mean growth.
View full question & answer→Question 271 Mark
If in the given data all the observations are of same values, then what will you say about the values of all the measures of central tendency?
AnswerIf in the given data all the observations are of same values, then the values of all the measures of central tendency are equal ‘to the value of observation.
View full question & answer→Question 281 Mark
If mean, median and mode are evenly distributed around the average, how are their values?
AnswerIf mean, median and mode are evenly distributed around the average, their values are same.
View full question & answer→Question 291 Mark
Which one is stable of the two measures - mode and geometric mean?
AnswerMode and geometric mean, of these two measures geometric mean is the stable measure.
View full question & answer→Question 301 Mark
In an inquiry of $100$ houses of a street, it is observed that the number of rooms in $58$ houses is $3$. In this data what is mode of number of rooms?
AnswerIn the data of number of rooms of $100$ houses, observation $3$ repeats $58$ times. Hence, the mode of number of room is $3$.
View full question & answer→Question 311 Mark
When can the value of mode be negative?
AnswerWhen the frequency distribution is highly skewed, the value of mode can be negative.
View full question & answer→Question 321 Mark
When is the empirical formula of mode used?
AnswerWhen the given frequency distribution is moderately skewed, or hi-modal or of unequal class length, then the approximate formula of mode is used.
View full question & answer→Question 331 Mark
Write the empirical formula of Karl Pearson for finding mode.
AnswerKarl Pearson's empirical formula for finding mode is as follows :
$Z=3 M-2 x$
where $Z=$ mode,$M=$ median, $\bar{x}=$ mean
View full question & answer→Question 341 Mark
What is an irregular continuous frequency distribution?
AnswerA frequency distribution in which classes are of unequal length and the fluctuations in the frequencies are not regular is called irregular continuous frequency distribution.
View full question & answer→Question 351 Mark
What is regular continuous frequency distribution?
AnswerA continuous frequency distribution in which the class length of all classes is equal and the fluctuations in the frequencies are equal.
View full question & answer→Question 361 Mark
Which type of observation is the mode in the data?
AnswerThe mode in the data is a observation which occurs most frequently.
View full question & answer→Question 371 Mark
In which fields is the mode widely used as a measure of average?
AnswerIn the fields of business, commerce and economical the mode is widely used measure of average.
View full question & answer→Question 381 Mark
Which measure of average is used to find the average of percentage increase or decrease in the value of variable?
AnswerTo find the average of percentage increase or decrease in the value of variable the geometric mean is used as a measure of average.
View full question & answer→Question 391 Mark
To represent the summary of qualitative data which is a suitable measure of average?
AnswerTo represent the summary of qualitative data median is a suitable measure of average.
View full question & answer→Question 401 Mark
Which measure of average is used in the construction of an index number?
AnswerIn the construction of an index number geometric mean is used as a measure of average.
View full question & answer→Question 411 Mark
State the mathematical relationship between the geometric mean and the arithmetic mean?
AnswerThe mathematical relationship between the geometric mean $(G)$ and the arithmetic $(A)$ is as follows:
$G \leq A$ means $A \geq G$
View full question & answer→Question 421 Mark
How is the value of geometric mean?
AnswerThe value of geometric mean is always positive.
View full question & answer→Question 431 Mark
State whether the following statement Is true or false:
AnswerIf it is false, rewrite the correct statement. “The value of highest observation of a data is called the mode.”
View full question & answer→Question 441 Mark
Which type of is the value of mode?
AnswerMode is the most common value of the data.
View full question & answer→Question 451 Mark
Can we find percentiles for an open end frequency distribution?
AnswerWe can find percentiles for an open end frequency distribution.
View full question & answer→Question 461 Mark
What is the difference between median and second quartile?
AnswerThere is no difference between median and second quartile.
$M = Q_2$
View full question & answer→Question 471 Mark
In a data $P_{25}=72$ and $Q_{3}=105$. What percentage of observations are more than $72$ and more than $105$ ?
Answer$P_{25}=72$ Hence, $100-25=75 \%$ of observations are more than $72 . Q_{3}=105$
Hence, $100-75=25 \%$ of observations are more than $105 .$
View full question & answer→Question 481 Mark
Find $P_{85}$ of observations $1.45,1.45,1.45,1.45,1.45$, $1.45,1.45,1.45$. All observations are of the same value $1.45$
AnswerHence, $P_{85}=1.45$.
View full question & answer→Question 491 Mark
$D_{3}$ of 20 observation is $20$ . If $2$ is subtracted from each observation and then divide by $3$, what will be the new value of $D_{3}$ ?
View full question & answer→Question 501 Mark
In a data of $15$ observations $D_{7}=32$, then can $Q_{3}$ be equal to $30 ?$
Answer$Q_{3} \geq D_{7}$. Hence, if $D_{7}=32, Q_{3}$ cannot be equal to $30 .$
View full question & answer→