Mix Examples - Statistics Questions in English

(C) We know the empirical relationship between Mean,Median,and Mode is given by: $Z = 3M - 2\bar{x}$.
Subtract $M$ from both sides of the equation:
$Z - M = 3M - 2\bar{x} - M$
$Z - M = 2M - 2\bar{x}$
Factor out $2$ on the right side:
$Z - M = 2(M - \bar{x})$
Divide both sides by $(M - \bar{x})$:
$\frac{Z - M}{M - \bar{x}} = 2$.

(B) We know the empirical relationship between mean,median,and mode is given by: $Z = 3M - 2\bar{x}$.
To find the value in the blank,we rearrange the equation:
$Z - \bar{x} = 3M - 2\bar{x} - \bar{x}$
$Z - \bar{x} = 3M - 3\bar{x}$
$Z - \bar{x} = 3(M - \bar{x})$
Comparing this with the given expression,the missing value is $3$.

(C) measure of central tendency is a summary statistic that represents the center point or typical value of a dataset. The three most common measures of central tendency are the Mean, Median, and Mode.
Range is a measure of dispersion (or variability), not central tendency. It is calculated as the difference between the maximum and minimum values in a dataset $(Range = \text{Maximum Value} - \text{Minimum Value})$.
Therefore, Range is not a measure of central tendency.

(D) First,calculate the total frequency $\Sigma f_{i} = 10 + 12 + 13 + 16 + 9 = 60$.
Thus,option $D$ is false because $\Sigma f_{i} = 60$,not $50$.
Now,let's verify other options:
$1$. Cumulative frequencies: $10, 22, 35, 51, 60$.
$2$. Median class: $N/2 = 60/2 = 30$. The cumulative frequency just greater than $30$ is $35$,which corresponds to the class $20-30$. So,option $A$ is true.
$3$. Modal class: The highest frequency is $16$,which corresponds to the class $30-40$. So,option $B$ is true.
$4$. Cumulative frequency of class $20-30$ is $10 + 12 + 13 = 35$. So,option $C$ is true.

(B) The empirical relationship between the three measures of central tendency (Mean,Median,and Mode) is given by the formula:
$Mode = 3 \times Median - 2 \times Mean$
In symbolic notation,this is represented as:
$Z = 3M - 2\bar{x}$
Where:
$Z$ = Mode
$M$ = Median
$\bar{x}$ = Mean
Therefore,the correct option is $B$.