Question

describing data with mean absolute deviation
a survey asked eight people about their wages and educational background. the table shows the hourly wages reported by people with and without a high school diploma.

no high school diploma	high school diploma
9.50	15.25
11.50	14.00
13.00	15.75

use the information to complete the statements.
the mean absolute deviation for people without a high school diploma is dropdown.
the mean absolute deviation for people with a high school diploma is dropdown.
the data for people without a high school diploma are more dropdown the mean than the data for people with a high school diploma.
dropdown options for the third statement: clustered around, spread out from

Explanation:

Part 1: Mean Absolute Deviation for No High School Diploma

Step 1: Calculate the mean

The data for no high school diploma: \(10.00, 9.50, 11.50, 13.00\)
Mean (\(\bar{x}\)) = \(\frac{10.00 + 9.50 + 11.50 + 13.00}{4}\) = \(\frac{44.00}{4}\) = \(11.00\)

Step 2: Calculate absolute deviations

\(\vert 10.00 - 11.00 \vert = 1.00\)
\(\vert 9.50 - 11.00 \vert = 1.50\)
\(\vert 11.50 - 11.00 \vert = 0.50\)
\(\vert 13.00 - 11.00 \vert = 2.00\)

Step 3: Calculate mean absolute deviation (MAD)

MAD = \(\frac{1.00 + 1.50 + 0.50 + 2.00}{4}\) = \(\frac{5.00}{4}\) = \(1.25\)

Part 2: Mean Absolute Deviation for High School Diploma

Step 1: Calculate the mean

The data for high school diploma: \(19.00, 15.25, 14.00, 15.75\)
Mean (\(\bar{x}\)) = \(\frac{19.00 + 15.25 + 14.00 + 15.75}{4}\) = \(\frac{64.00}{4}\) = \(16.00\)

Step 2: Calculate absolute deviations

\(\vert 19.00 - 16.00 \vert = 3.00\)
\(\vert 15.25 - 16.00 \vert = 0.75\)
\(\vert 14.00 - 16.00 \vert = 2.00\)
\(\vert 15.75 - 16.00 \vert = 0.25\)

Step 3: Calculate mean absolute deviation (MAD)

MAD = \(\frac{3.00 + 0.75 + 2.00 + 0.25}{4}\) = \(\frac{6.00}{4}\) = \(1.50\) Wait, no, wait: \(3 + 0.75 = 3.75\), \(3.75 + 2 = 5.75\), \(5.75 + 0.25 = 6\). Then \(6/4 = 1.5\)? Wait, but let's recalculate:

Wait, 19 - 16 = 3; 16 - 15.25 = 0.75; 16 - 14 = 2; 16 - 15.75 = 0.25. Sum: 3 + 0.75 + 2 + 0.25 = 6. Then 6/4 = 1.5. Wait, but earlier for no diploma, MAD was 1.25. Wait, but let's check the third part: "The data for people without a high school diploma are more [clustered around / spread out from] the mean than the data for people with a high school diploma." Wait, lower MAD means more clustered. Wait, no: MAD for no diploma is 1.25, MAD for diploma is 1.5. So lower MAD means more clustered. Wait, but the options are "clustered around" or "spread out from". Wait, maybe I made a mistake.

Wait, let's recalculate MAD for high school:

Data: 19, 15.25, 14, 15.75. Mean is (19 + 15.25 +14 +15.75)/4 = (19 + 15.25 is 34.25; 14 +15.75 is 29.75; total 34.25 +29.75 = 64; 64/4=16. Correct.

Absolute deviations:

|19 -16| = 3

|15.25 -16| = 0.75

|14 -16| = 2

|15.75 -16| = 0.25

Sum: 3 + 0.75 = 3.75; +2 = 5.75; +0.25 = 6. 6/4 = 1.5. Correct.

For no diploma:

Data: 10, 9.5, 11.5, 13. Mean is (10 +9.5 +11.5 +13)/4 = (10+9.5=19.5; 11.5+13=24.5; total 19.5+24.5=44; 44/4=11. Correct.

Absolute deviations:

|10-11|=1

|9.5-11|=1.5

|11.5-11|=0.5

|13-11|=2

Sum: 1 +1.5=2.5; +0.5=3; +2=5. 5/4=1.25. Correct.

So MAD for no diploma is 1.25, MAD for diploma is 1.5. So lower MAD means more clustered. So the data for no diploma are more clustered around the mean than those with diploma.

Wait, but the question says "The data for people without a high school diploma are more [clustered around / spread out from] the mean than the data for people with a high school diploma." So since MAD is lower (1.25 < 1.5), they are more clustered around the mean.

So:

1. The mean absolute deviation for people without a high school diploma is \(1.25\).

1. The mean absolute deviation for people with a high school diploma is \(1.50\).

1. The data for people without a high school diploma are more clustered around the mean than the data for people with a high school diploma.

Answer:

s:

1. \(1.25\)

1. \(1.50\)

1. clustered around