Question

Write short notes

1) Merits and demerits of mode.

2) Write merits and demerits of median.

3) Define quartile- Give the formula for computing (1) Q₁, (2) Q₂, (3) Q₃

4) Define deciles. Give the formulas for computing (1) D₅, (ii) D₇ (iii) D₉

5) Define percentiles. Give the formulas. For computing the 3^rd percentiles and 99^th percentiles.

6) Give definitions

(1) Median (2) Mode

Answer 1

1. The following are the merits and demerits of the mode:

Merits:

i. It is easy to calculate.
ii. It is not influenced by extreme values.
iii. It is useful in the analysis of comprehensive data.
iv. It can be graphically determined.
v. Its value can be measured in open-ended class intervals as well.
vi. It can be determined or measured by inspection or location only.

Demerits:

i. The value of mode is ambiguous and indefinite in the sense that a series can even have two modes.
ii. It is not based on all observations.
iii. It is a mere positional average rather than arithmetic average.
iv. Its scope is limited in small samples.
v. It is an unstable average.

2. The following are the merits and demerits of the median:

Merits:

i. It is easy and simple to calculate.
ii. It is not influenced by extreme values.
iii. All details of data are not required in the calculation of median.
iv. It can be determined by mere inspection and graphically as well.
v. It is considered an ideal measure for studying qualitative attributes.

Demerits:

i. It is not based on all observations.
ii. It is not suitable for further algebraic treatment.
iii. It is not a true representative of data if the data lack uniformity.
iv. It is greatly affected by sampling fluctuations.
v. It ignores the extreme values of data.

3. Quartiles are the values of a given set of data that divide the whole data or set of observations into four equal parts. The following are the formulas for computing the three quartiles:

$Q_1={\left(\frac{N+1}{4}\right)}^{th}observation$

$Q_2=2{\left(\frac{N+1}{4}\right)}^{th}observation$

$Q_3=3{\left(\frac{N+1}{4}\right)}^{th}observation$

4. Deciles are the values of a given set of data that divide the whole data or set of observations into 10 equal parts. There are 9 deciles in data that divide it into 10 parts of the same size, namely $D_1{D}_2{D}_3{D}_4{D}_5{D}_6{D}_7{D}_8{D}_9$.
The following are the formulas for D5, D7 and D9:

$D_5=5{\left(\frac{N+1}{10}\right)}^{th}item$

$D_7=7{\left(\frac{N+1}{10}\right)}^{th}item$

$D_9=9{\left(\frac{N+1}{9}\right)}^{th}item$

5. Percentiles are the values of a given set of data that divide the whole set of observations into 100 equal parts. There are 99 percentiles that give 99 dividing points denoted by $P_1{P}_2{P}_3...........P_{99}$.
The following are the formulas for computing the 3rd and 99th percentiles:

$$\begin{gathered} P_3=3{\left(\frac{N+1}{100}\right)}^{th}item \\ {P}_{99}=99{\left(\frac{N+1}{100}\right)}^{th}item \end{gathered}$$

6. (1) The median is the middlemost value of a given set of data arranged in an array (ascending or descending). It divides the data into two equal groups. Mathematically, for an individual series, it is calculated as:
M = Size of the ${\left(\frac{N+1}{2}\right)}^{th}$ item
Here,
M = Median
N = Total number of items

(2) The mode is the observation or value that repeats itself the maximum number of times or that has the maximum frequency. It is easily measured by merely looking and locating the value with the highest frequency.