Question

the bar graph summarizes the household size of a social worker’s client families. which statement best describes the relationship between the mean and median household size? choose 1 answer: a the mean is greater than the median. b the median is greater than the mean. c the mean and the median are equal. d there is not enough information to compare the mean and the median.

Explanation:

Step1: Calculate total number of families

First, we find the total number of client families by adding the number of families for each household size. From the bar graph:

• Household size 2: 2 families
• Household size 3: 1 family
• Household size 4: 16 families
• Household size 5: 7 families
• Household size 6: 2 families
• Household size 7: 1 family
• Household size 8: 1 family

Total number of families \( n = 2 + 1 + 16 + 7 + 2 + 1 + 1 = 30 \)

Step2: Find the median

Since \( n = 30 \) (even), the median is the average of the \( \frac{n}{2} = 15 \)-th and \( \frac{n}{2}+1 = 16 \)-th values when the data is ordered.

Let's list the cumulative frequencies:

• Household size 2: cumulative frequency = 2 (covers positions 1 - 2)
• Household size 3: cumulative frequency = 2 + 1 = 3 (covers positions 3 - 3)
• Household size 4: cumulative frequency = 3 + 16 = 19 (covers positions 4 - 19)

So the 15th and 16th values are both in the household size 4 group. Thus, the median \( M = \frac{4 + 4}{2} = 4 \)

Step3: Calculate the mean

The mean \( \bar{x} \) is calculated as \( \bar{x} = \frac{\sum (x_i \cdot f_i)}{\sum f_i} \), where \( x_i \) is the household size and \( f_i \) is the number of families with that size.

\[

$$\begin{align*} \sum (x_i \cdot f_i) &= (2 \cdot 2) + (3 \cdot 1) + (4 \cdot 16) + (5 \cdot 7) + (6 \cdot 2) + (7 \cdot 1) + (8 \cdot 1) \\ &= 4 + 3 + 64 + 35 + 12 + 7 + 8 \\ &= 4 + 3 = 7; \quad 7 + 64 = 71; \quad 71 + 35 = 106; \quad 106 + 12 = 118; \quad 118 + 7 = 125; \quad 125 + 8 = 133 \end{align*}$$

\]

Since \( \sum f_i = 30 \), the mean \( \bar{x} = \frac{133}{30} \approx 4.43 \)

Step4: Compare mean and median

We have mean \( \approx 4.43 \) and median \( = 4 \). So the mean is greater than the median.

Answer:

A. The mean is greater than the median.