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Difficulty: HardDescriptive Statistics and Data Representations

A geology student collected a group of rocks and recorded their weights, in grams, in the table below. One frequency, ff, is missing.

Weight (grams)Frequency
255
104
20ff
302
156

If the mean weight of the rocks in the group is exactly 18.7518.75 grams, what is the median weight, in grams, of the rocks?

  1. A
    3
  2. B
    10
  3. C
    15
  4. 17.5Answer
  5. E
    20

Answer

17.5
The correct answer is 17.5. By setting up the weighted mean equation, we find that the missing frequency ff is 3. This means there are 20 rocks in total. When sorted in ascending order, the weights are four 10s, six 15s, three 20s, five 25s, and two 30s. The 10th rock weighs 15 grams and the 11th rock weighs 20 grams. The median is the average of these two middle values: (15 + 20) / 2 = 17.5 grams.

Step-by-Step Solution

1
Set up an equation for the mean weight using the frequencies and weights from the table.
The sum of the weights is 25(5)+10(4)+20(f)+30(2)+15(6)=315+20f25(5) + 10(4) + 20(f) + 30(2) + 15(6) = 315 + 20f. The total number of rocks is 5+4+f+2+6=17+f5 + 4 + f + 2 + 6 = 17 + f. The mean is 315+20f17+f=18.75\frac{315 + 20f}{17 + f} = 18.75.
To find the missing frequency ff using the given mean.
2
Solve the equation for ff.
315+20f=18.75(17+f)  ⟹  315+20f=318.75+18.75f  ⟹  1.25f=3.75  ⟹  f=3315 + 20f = 18.75(17 + f) \implies 315 + 20f = 318.75 + 18.75f \implies 1.25f = 3.75 \implies f = 3.
To determine the number of rocks that weigh 20 grams.
3
Find the total number of rocks and identify the median position.
The total number of rocks is 17+3=2017 + 3 = 20. Since the total number is even, the median is the average of the 10th and 11th values when the weights are sorted in ascending order.
The median of an even number of data points is the average of the two middle values.
4
List the sorted weights and find the 10th and 11th values.
The sorted weights are: four 10s, six 15s (making 10 rocks total), followed by three 20s. Thus, the 10th rock weighs 15 grams and the 11th rock weighs 20 grams.
To find the values of the two middle rocks.
5
Calculate the average of the 10th and 11th values.
Median = 15+202=17.5\frac{15 + 20}{2} = 17.5 grams.
To find the median weight.

Key Concept

Calculating the median from a frequency distribution table after finding a missing frequency using the mean.
Estimated Time:2m 30s
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Topics

Descriptive Statistics and Data Representations

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