Violet creates two spinners for a game. Each spinner is spun once, and the sum is recorded. The table represents the sums of the spinners and the frequency of each sum.

Possible Sums
\begin{tabular}{|c|c|}
\hline Sum & Frequency \\
\hline 5 & 1 \\
\hline 7 & 2 \\
\hline 9 & 3 \\
\hline 11 & 4 \\
\hline 13 & 3 \\
\hline 15 & 2 \\
\hline 17 & 1 \\
\hline
\end{tabular}

What statement is true about the mean of the sums of the two spinners?

A. The mean is 12.
B. The mean is 16.
C. The mean is the same as the median.
D. The mean is the same as the range.



Answer :

To determine the correct statement about the mean of the sums of the two spinners, let's analyze and calculate the mean, median, and range step by step.

### Step 1: Calculate the Mean

The mean (or average) is calculated by taking the sum of each possible sum multiplied by its frequency, and then dividing by the total frequency.

Sums and Frequencies:
- Sum = 5, Frequency = 1
- Sum = 7, Frequency = 2
- Sum = 9, Frequency = 3
- Sum = 11, Frequency = 4
- Sum = 13, Frequency = 3
- Sum = 15, Frequency = 2
- Sum = 17, Frequency = 1

Total sum:
[tex]\[ \sum_{i=1}^{7} (\text{sum}_i \times \text{frequency}_i) = (5 \times 1) + (7 \times 2) + (9 \times 3) + (11 \times 4) + (13 \times 3) + (15 \times 2) + (17 \times 1) = 5 + 14 + 27 + 44 + 39 + 30 + 17 = 176 \][/tex]

Total frequency:
[tex]\[ \sum_{i=1}^{7} \text{frequency}_i = 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 \][/tex]

Mean:
[tex]\[ \text{Mean} = \frac{\text{Total sum}}{\text{Total frequency}} = \frac{176}{16} = 11.0 \][/tex]

### Step 2: Calculate the Median

The median is the middle value of a data set ordered from least to greatest.
To find the median, extend each sum by its frequency.

Ordered data set:
[tex]\[ \{5, 7, 7, 9, 9, 9, 11, 11, 11, 11, 13, 13, 13, 15, 15, 17\} \][/tex]

Since there are 16 values (an even number), the median is the average of the 8th and 9th values in the ordered data.
8th value is 11 and 9th value is also 11.

Median:
[tex]\[ \text{Median} = \frac{11 + 11}{2} = 11.0 \][/tex]

### Step 3: Calculate the Range

The range is the difference between the maximum and minimum values of the sums.

Minimum sum = 5
Maximum sum = 17

Range:
[tex]\[ \text{Range} = 17 - 5 = 12 \][/tex]

### Conclusion

Based on the calculations:

- The mean is 11.0
- The median is 11.0
- The range is 12

The correct statement from the options is:
[tex]\[ \textbf{The mean is the same as the median.} \][/tex]