Question

the monthly incomes of a group of 8 individuals are as follows: $3,000, $3,500, $3,600, $3,700, $3,800, $4,000, $4,500, $12,000. the mean income for this group is $4,762.50, while the median income is $3,750. based on the comparison of the mean and median incomes, which statement best describes the income distribution in the context of this scenario?
there is an individual or a few individuals earning substantially more than the others, causing the mean to be higher than the median.
the incomes are uniformly distributed, with each individual earning around the same amount.
there is an individual or a few individuals earning significantly less than the others, causing the mean to be lower than the median.
most of the individuals have incomes close to the mean of $4,762.50.

Explanation:

Step1: Arrange the data in ascending order

The incomes are: \$3,000, \$3,500, \$3,600, \$3,700, \$3,800, \$4,000, \$4,500, \$12,000

Step2: Calculate the median

For \(n = 8\) (even number of data points), the median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+ 1)\)th values. \(\frac{n}{2}=4\) and \(\frac{n}{2}+1 = 5\). The 4th value is \$3,700 and the 5th value is \$3,800. Median \(M=\frac{3700 + 3800}{2}=\$3,750\)

Step3: Calculate the mean

Mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\). \(\sum_{i=1}^{8}x_{i}=3000+3500 + 3600+3700+3800+4000+4500+12000=\$38,100\). Mean \(\bar{x}=\frac{38100}{8}=\$4,762.5\)

Step4: Analyze the effect of the out - lier

The value \(\$12,000\) is an out - lier. The mean is affected by extreme values (out - liers), while the median is more resistant. Since there is a relatively large out - lier (\(\$12,000\)) pulling the mean up.

Answer:

There is an individual or a few individuals earning substantially more than the others, causing the mean to be higher than the median.