Questions: Find the deviations from the mean for the set of ACT test score data: 22, 25, 18, 20, 20. (Simplify your answers.) Score Deviation from mean 22 25 18 20 20

Find the deviations from the mean for the set of ACT test score data: 22, 25, 18, 20, 20.

(Simplify your answers.)
Score Deviation from mean
22

25

18

20

20

Transcript text: Find the deviations from the mean for the set of ACT test score data: $22,25,18,20,20$. (Simplify your answers.) Score Deviation from mean 22 $\square$ 25 $\square$ 18 $\square$ 20 $\square$ 20 $\square$

Solution

Solution Steps

Step 1: Calculate the Mean

To find the mean ($ \mu $) of the ACT test scores $ 22, 25, 18, 20, 20 $, we use the formula:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{22 + 25 + 18 + 20 + 20}{5} = \frac{105}{5} = 21.0 \]

Thus, the mean of the data is $ \mu = 21.0 $.

Step 2: Calculate Deviations from the Mean

Next, we calculate the deviations from the mean for each score. The deviation for each score $ x_i $ is given by:

\[ \text{Deviation} = x_i - \mu \]

Calculating the deviations:

For $ 22 $: $ 22 - 21.0 = 1.0 $
For $ 25 $: $ 25 - 21.0 = 4.0 $
For $ 18 $: $ 18 - 21.0 = -3.0 $
For $ 20 $: $ 20 - 21.0 = -1.0 $
For $ 20 $: $ 20 - 21.0 = -1.0 $

The deviations from the mean are as follows:

Score: $ 22 $, Deviation: $ 1.0 $
Score: $ 25 $, Deviation: $ 4.0 $
Score: $ 18 $, Deviation: $ -3.0 $
Score: $ 20 $, Deviation: $ -1.0 $
Score: $ 20 $, Deviation: $ -1.0 $

Final Answer

The mean of the ACT test scores is $ \mu = 21.0 $ and the deviations from the mean are:

For score $ 22 $: $ 1.0 $
For score $ 25 $: $ 4.0 $
For score $ 18 $: $ -3.0 $
For score $ 20 $: $ -1.0 $
For score $ 20 $: $ -1.0 $

Thus, the final answer is:

\[ \boxed{\text{Mean} = 21.0, \text{Deviations} = \{1.0, 4.0, -3.0, -1.0, -1.0\}} \]

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