Question

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.

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

the mean is 12.
the mean is 16.
the mean is the same as the median.
the mean is the same as the range.

Explanation:

Step1: Calculate the total number of data points

$$ 1 + 2+3 + 4+3 + 2+1=\sum_{i = 1}^{7}f_i=16 $$

Step2: Calculate the sum of all values

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Step3: Calculate the mean

The mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_if_i}{\sum_{i = 1}^{n}f_i}\), so \(\bar{x}=\frac{176}{16} = 11\)

Step4: Find the median

Since \(n = 16\) (even), the median is the average of the \(8^{th}\) and \(9^{th}\) values.
The cumulative frequencies:

• For sum \(5\): cumulative frequency \(=1\)
• For sum \(7\): cumulative frequency \(=1 + 2=3\)
• For sum \(9\): cumulative frequency \(=3+3 = 6\)
• For sum \(11\): cumulative frequency \(=6 + 4=10\)

The \(8^{th}\) and \(9^{th}\) values are both \(11\), so the median \(=\frac{11 + 11}{2}=11\)

Step5: Calculate the range

The range \(=17-5 = 12\)

Answer:

The mean is the same as the median.