Mean and Median of M&M Weights: Histogram Analysis

Math Problem Statement

The accompanying data represent the weights (in grams) of a simple random sample of

4848

M&M plain candies. Determine the shape of the distribution of weights of M&Ms by drawing a frequency histogram. Find the mean and median. Which measure of central tendency better describes the weight of a plain M&M?

Click the icon to view the candy weight data.

Question content area bottom

Part 1

Draw a frequency histogram. Choose the correct graph below.

Weight of Plain M and Ms

0.780.840.90.960612Weight (grams)Frequency

A histogram titled Weight of Plain M and M's has a horizontal axis labeled Weight in grams from 0.78 to 0.96 in increments of 0.02 and a vertical axis labeled Frequency from 0 to 12 in increments of 1. The histogram has vertical bars of width 0.02 starting at the horizontal axis value 0.78. The approximate heights of the bars are as follows, where the left horizontal axis label is listed first and the approximate height is listed second: 0.78, 1; 0.8, 1; 0.82, 9; 0.84, 10; 0.86, 9; 0.88, 8; 0.9, 4; 0.92, 2; 0.94, 4.

Weight of Plain M and Ms

0.780.840.90.960612Weight (grams)Frequency

A histogram titled Weight of Plain M and M's has a horizontal axis labeled Weight in grams from 0.78 to 0.96 in increments of 0.02 and a vertical axis labeled Frequency from 0 to 12 in increments of 1. The histogram has vertical bars of width 0.02 starting at the horizontal axis value 0.78. The approximate heights of the bars are as follows, where the left horizontal axis label is listed first and the approximate height is listed second: 0.78, 1; 0.8, 9; 0.82, 10; 0.84, 8; 0.86, 9; 0.88, 4; 0.9, 4; 0.92, 1; 0.94, 2.

Your answer is not correct.

Weight of Plain M and Ms

0.780.840.90.960612Weight (grams)Frequency

A histogram titled Weight of Plain M and M's has a horizontal axis labeled Weight in grams from 0.78 to 0.96 in increments of 0.02 and a vertical axis labeled Frequency from 0 to 12 in increments of 1. The histogram has vertical bars of width 0.02 starting at the horizontal axis value 0.78. The approximate heights of the bars are as follows, where the left horizontal axis label is listed first and the approximate height is listed second: 0.78, 1; 0.8, 1; 0.82, 4; 0.84, 9; 0.86, 4; 0.88, 10; 0.9, 8; 0.92, 9; 0.94, 2.

Weight of Plain M and Ms

0.780.840.90.960612Weight (grams)Frequency

A histogram titled Weight of Plain M and M's has a horizontal axis labeled Weight in grams from 0.78 to 0.96 in increments of 0.02 and a vertical axis labeled Frequency from 0 to 12 in increments of 1. The histogram has vertical bars of width 0.02 starting at the horizontal axis value 0.78. The approximate heights of the bars are as follows, where the left horizontal axis label is listed first and the approximate height is listed second: 0.78, 1; 0.8, 1; 0.82, 4; 0.84, 8; 0.86, 9; 0.88, 10; 0.9, 9; 0.92, 4; 0.94, 2.

This is the correct answer.

Part 2

The mean is

x overbarxequals=0.8800.880

gram(s).

Solution

Part 1: Frequency Histogram Shape

From the provided descriptions, the histogram that best represents the weight distribution of plain M&M's appears to be D, as it shows a distribution with the most frequency in the middle and tapering off at both ends. This suggests a roughly symmetrical shape with a peak around the middle values (around 0.86–0.88 grams).

Part 2: Mean and Median

The mean of the data is given as $\bar{x} = 0.880 \, \text{grams}$ .
Based on the histogram shape, the median would likely be close to the mean, since the distribution looks symmetrical. Without exact data points, it's hard to determine the precise median, but it should be approximately $0.88 \, \text{grams}$ .

Which Measure of Central Tendency is Better?

For this data:

Since the distribution appears symmetrical, the mean and median are likely to be quite close. In cases of symmetry, the mean generally provides a better description of the central tendency because it takes into account all data points.

Thus, the mean better describes the weight of a plain M&M, given the symmetrical nature of the histogram.

Would you like more details about the histogram, the calculations, or the reasoning behind this conclusion?

Here are some related questions:

How do you calculate the mean and median of a data set?
What is the difference between symmetric and skewed distributions?
How does the shape of a histogram impact the choice of central tendency measure?
How do you identify the shape of a histogram just by looking at the bars?
What would happen to the mean and median if the distribution were skewed?

Tip: Always consider the shape of the distribution when choosing between the mean and median. If the data is skewed, the median is often a better measure because it is less affected by extreme values.

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Math Problem Analysis

Mathematical Concepts

Statistics
Descriptive Statistics
Central Tendency
Histograms

Formulas

Mean formula: $ \bar{x} = \frac{\sum x_i}{n} $
Median: the middle value of a data set

Theorems

Symmetry in distributions
Properties of central tendency measures in symmetrical distributions

Suitable Grade Level

Grades 9-12

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