Consider the distribution below. a) From the distribution, obtain the five number summary. b) What is the (i) mode and the (ii) midhinge of the distribution? c) Calculate the mean... Consider the distribution below. a) From the distribution, obtain the five number summary. b) What is the (i) mode and the (ii) midhinge of the distribution? c) Calculate the mean using the relationship existing among mode, median, and mean. Read more

Steps to Solve

1. Listing the values in ascending order

First, we need to arrange the provided data in ascending order:

$$ 1, 2, 2, 2, 3, 4, 4, 5, 5, 6, 6, 7, 8, 8, 10, 14, 14, 15, 16, 16, 38, 45, 50, 478, 3555678899, 123445678899, 1234455556789 $$

2. Calculating the Five Number Summary

The five number summary includes the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum.

• Minimum: The smallest number is $1$.

• Maximum: The largest number is $1234455556789$.

• Median (Q2): As there are 27 data points, the median is the value at position $\frac{27 + 1}{2} = 14$. The 14th value in sorted order is $8$.

• First Quartile (Q1): The median of the first half of the data (the first 13 values) is the 7th value, which is $4$.

• Third Quartile (Q3): The median of the second half of the data (the last 13 values) is the 20th value, which is $16$.

So, the five number summary is:

• Min: $1$
• Q1: $4$
• Median (Q2): $8$
• Q3: $16$
• Max: $1234455556789$

3. Finding the Mode

Mode is the number that appears most frequently. By examining the data, the number $2$ occurs three times, making it the mode.

4. Calculating the Midhinge

Midhinge is calculated using the formula:

$$ \text{Midhinge} = \frac{Q1 + Q3}{2} $$

Substituting the values:

$$ \text{Midhinge} = \frac{4 + 16}{2} = 10 $$

5. Calculating the Mean

The relationship among mode, median, and mean is given as:

$$ \text{Mean} = \frac{\text{Mode} + 3\cdot\text{Median}}{4} $$

Substituting calculated values:

$$ \text{Mean} = \frac{2 + 3 \cdot 8}{4} = \frac{2 + 24}{4} = \frac{26}{4} = 6.5 $$