Question

Difficulty: MediumDescriptive Statistics and Data Representations

A high school science club organized a daily recycling drive over a 10-day period. The frequency table below summarizes the number of aluminum cans collected per day:

Cans Collected Per DayNumber of Days
102
203
303
401
501

What is the positive difference between the mean number of cans collected per day and the median number of cans collected per day?

  1. A
    0
  2. 1Answer
  3. C
    4
  4. D
    5
  5. E
    6

Answer

The positive difference between the mean and median is 1.
To find the mean, calculate the weighted sum of the products of each daily quantity and its frequency: (10 × 2 + 20 × 3 + 30 × 3 + 40 × 1 + 50 × 1) = 260. Dividing 260 by the total 10 days gives a mean of 26. To find the median, list the 10 data points in order: 10, 10, 20, 20, 20, 30, 30, 30, 40, 50. The middle two numbers are the 5th and 6th terms (20 and 30), so the median is (20 + 30) / 2 = 25. Subtracting 25 from 26 yields a positive difference of 1.

Step-by-Step Solution

1
Calculate the total number of cans collected and the total number of days.
Total days = 2 + 3 + 3 + 1 + 1 = 10. Total cans = (10 × 2) + (20 × 3) + (30 × 3) + (40 × 1) + (50 × 1) = 20 + 60 + 90 + 40 + 50 = 260.
To find the mean from a frequency table, multiply each data value by its frequency and sum the results, then divide by the total frequency.
2
Calculate the mean number of cans collected per day.
Mean = 260 / 10 = 26.
Divide the total sum of collected cans by the total number of days.
3
Find the median of the data set.
The 10 ordered data points are: 10, 10, 20, 20, 20, 30, 30, 30, 40, 50. The 5th value is 20 and the 6th value is 30. Median = (20 + 30) / 2 = 25.
For an even number of data points (N=10N = 10), the median is the arithmetic mean of the two middle values (N/2N/2 th and (N/2+1)(N/2 + 1) th terms).
4
Compute the positive difference between the mean and median.
Difference = 26 - 25 = 1.
Subtract the median from the mean to find the requested difference.

Key Concept

Weighted Mean and Median from Frequency Tables
Estimated Time:1m 15s
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