The daily electricity consumption, in kilowatt-hours (kWh), of a household was recorded fo… — SAT

The daily electricity consumption, in kilowatt-hours (kWh), of a household was recorded for 7 consecutive days: $12$ , $12$ , $13$ , $14$ , $14$ , $15$ , and $45$ . If the household's electricity consumption of $45\text{ kWh}$ on the seventh day is removed from the dataset, which of the following statements best describes the effect on the mean and the median of the dataset?

The mean decreases by a greater amount than the median.Answer
B
The median decreases by a greater amount than the mean.
C
The mean decreases, but the median increases.
D
The mean increases, but the median decreases.

Answer

The mean decreases by a greater amount than the median.

The correct answer is correct because calculating the original mean yields

\approx 17.86\text{ kWh}

and the original median yields

14\text{ kWh}

. After removing the outlier (

45\text{ kWh}

), the mean decreases to

\approx 13.33\text{ kWh}

(a change of

4.53\text{ kWh}

) and the median decreases to

13.5\text{ kWh}

(a change of

0.5\text{ kWh}

). Comparing these changes shows the mean decreased by a larger amount.

Step-by-Step Solution

Calculate the mean and median of the original 7-day dataset.

The original mean is

\frac{12 + 12 + 13 + 14 + 14 + 15 + 45}{7} = \frac{125}{7} \approx 17.86\text{ kWh}

. The sorted dataset is

12, 12, 13, 14, 14, 15, 45

, so the median is the middle value, which is

14\text{ kWh}

This establishes baseline values for the two measures.

Calculate the mean and median after removing the outlier of

45\text{ kWh}

The new mean is

\frac{12 + 12 + 13 + 14 + 14 + 15}{6} = \frac{80}{6} \approx 13.33\text{ kWh}

. The new sorted dataset is

12, 12, 13, 14, 14, 15

, so the median is the average of the two middle values:

\frac{13 + 14}{2} = 13.5\text{ kWh}

This provides the new values for comparison.

Compare the changes in the mean and the median.

The mean decreased by

17.86 - 13.33 = 4.53\text{ kWh}

. The median decreased by

14 - 13.5 = 0.5\text{ kWh}

. The decrease in the mean is greater.

This confirms the statement that describes the relative changes.

Key Concept

Effect of outliers on mean and median

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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