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.Cevap
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.

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Effect of outliers on mean and median

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru