A customer satisfaction survey asked 25 patrons at a local restaurant to rate their dining… — ACT

A customer satisfaction survey asked $25$ patrons at a local restaurant to rate their dining experience on a scale from $1$ (poor) to $5$ (excellent). The frequency table below summarizes the ratings collected:

Rating	Number of Patrons
1	3
2	5
3	8
4	6
5	3

What is the positive difference between the mean rating and the median rating of these $25$ patrons?

$0.04$ Answer
B
$0.00$
C
$1.96$
D
$12.20$
E
$58.60$

Answer

The positive difference between the mean rating and the median rating is

0.04

To find the positive difference, first compute the mean by calculating the sum of products of each rating and its frequency:

(1 \times 3) + (2 \times 5) + (3 \times 8) + (4 \times 6) + (5 \times 3) = 76

. Dividing

76

by the total number of patrons (

25

) yields a mean of

3.04

. Next, find the median by identifying the

13\text{th}

ordered rating out of

25

. The cumulative frequency reaches

8

after rating 2 and

16

after rating 3, so the

13\text{th}

entry is

3

. The positive difference between the mean (

3.04

) and median (

3.00

) is

0.04

Step-by-Step Solution

Calculate the total sum of all ratings.

Sum

= (1 \times 3) + (2 \times 5) + (3 \times 8) + (4 \times 6) + (5 \times 3) = 3 + 10 + 24 + 24 + 15 = 76

Each rating value must be multiplied by its frequency to find the total value of all responses.

Calculate the mean rating.

Mean

= \frac{76}{25} = 3.04

Divide the total sum of all ratings by the total number of patrons (

25

Determine the median rating.

Median

= 3

With

25

data points arranged in ascending order, the median is the

\frac{25 + 1}{2} = 13\text{th}

score. Accumulating frequencies: Ratings 1 and 2 cover the first

3 + 5 = 8

scores, while Rating 3 covers the

9\text{th}

through

16\text{th}

scores. Thus, the

13\text{th}

score is

3

Find the positive difference between the mean and median.

Difference

= 3.04 - 3.00 = 0.04

Subtract the median rating (

3

) from the mean rating (

3.04

Key Concept

Calculating Mean and Median from a Frequency Table

Estimated Time:1m 30s

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