Question

The table below shows discrete frequency distribution data. Use it to answer the questions that follow.
Class 0-4 5-9 10-14 15-19 20-24 25-29 30-34 35-39
Frequency 5 8 10 12 7 6 3 2

Compute;
(i) Mode of the distribution
(ii) The 7th decile
(iii)The third quartile

Answer 1

Answer on Question #63551 – Math – Statistics and Probability Question

The table below shows discrete frequency distribution data. Use it to answer the questions that follow.

Compute:

(i) Mode of the distribution

(ii) The 7th decile

(iii) The third quartile

Solution

(i) To find the mode of grouped distribution, the following formula will be used:

$$Mode = l + \frac {f _ {1} - f _ {0}}{2 f _ {1} - f _ {0} - f _ {2}} \cdot h,$$

where $l$ is the lower limit of model class, $f_{0}$ is the frequency of class preceding, $f_{1}$ is the frequency of that class and $f_{2}$ is the frequency of class succeeding the model class respectively, $h$ is the class width.

Let's put the numbers into a table:

The mode containing class is [15-19] has the biggest frequency 12.

So the mode value is

$$Mode = 15 + \frac {12 - 10}{2 \cdot 12 - 10 - 7} \cdot 4 = 16 \frac {1}{7} \approx 16.14.$$

(ii) To find the 7th decile, we need to use the formula:

$$D _ {k} = l _ {i} + \frac {\frac {k}{10} \cdot \sum f - f _ {D _ {k} - 1} ^ {\prime}}{f _ {D _ {k} - 1}} \cdot h,$$

where $l_{i}$ is the lower limit of decile class, $\sum f$ is the sum of the absolute frequency; $f_{D_{k-1}}'$ is absolute frequency lies below the decile class; $f_{D_{k-1}}$ is frequency of the decile class; $k$ is the decile number; $h$ is the class width.

The 7th decile containing class is [20-24], because Cumulative frequency in that interval is $42 > 37.1 = \frac{53}{10} \cdot 7$.

Therefore

$$D_k = 20 + \frac{\frac{7}{10} \cdot 53 - 35}{7} \cdot 4 = 21.2.$$

(iii) To find the third quartile, we need to use the formula:

$$Q_3 = l + \frac{0.75 \cdot \sum f - f_{Q_{3-1}}'}{f_{Q_{3-1}}} \cdot h,$$

where $l$ is the lower limit of the third quartile class, $\sum f$ is the sum of the absolute frequency; $f_{Q_{3-1}}'$ is absolute frequency lies below the quartile class; $f_{Q_{3-1}}$ is frequency of the quartile class; $h$ is the class width.

The third quartile containing class is [20-24], because Cumulative frequency in that interval is $42 > 39.75 = \frac{53}{4} \cdot 3$.

$$Q_3 = 20 + \frac{0.75 \cdot 53 - 35}{7} \cdot 4 = 22.71.$$

Answer: (i) 16.14; (ii) 21.2; (iii) 22.71.

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