Comparing Two Numerical Data Distributions Using Dot Plots and Box Plots - Seventh Grade Mathematics
Comparing numerical distributions requires synthesizing measures of center and variability, evaluating graphical overlap, and quantifying separation relative to dispersion.
The Comparative Framework: Center and Spread
In seventh grade statistics, comparing two distinct populations or experimental groups requires analyzing two structural dimensions simultaneously:
1. Measure of Center: Where is the typical middle of each distribution located? (Mean or Median)
2. Measure of Spread: How dispersed and variable is each distribution? (MAD or Interquartile Range)
MATCHING CENTER AND SPREAD
Symmetric Distributions: Compare MEANS and MADS
Skewed Distributions: Compare MEDIANS and IQRS
Always pair the mean with the Mean Absolute Deviation (MAD), and pair the median with the Interquartile Range (IQR). Mixing measures (such as comparing a mean to an IQR) undermines statistical validity.
Comparing Distributions with Visual Dot Plots
A dot plot displays individual data observations as stacked dots above a calibrated number line, allowing direct visual comparison of cluster shapes, peaks, and spread.
HOURS OF SLEEP PER NIGHT:
Seventh Graders:
*
* * *
* * * * *
* * * * * *
<----+----+----+----+----+----+----+----+---->
6 7 8 9 10 11 12
Center: Median = 8.5 hours. Spread: Range = 11 - 6 = 5 hours.
High School Seniors:
*
* *
* * * *
* * * * *
<----+----+----+----+----+----+----+----+---->
5 6 7 8 9 10 11
Center: Median = 6.5 hours. Spread: Range = 10 - 5 = 5 hours.
By placing the two dot plots on the identical horizontal scale, we observe that while both groups have comparable overall spread (five hours), the seventh-grade distribution is shifted approximately two hours to the right of the high school distribution.
Comparing Distributions with Parallel Box Plots
Parallel box plots display two or more box-and-whisker plots stacked vertically along the same coordinate axis, enabling instantaneous visual comparison of five-number summaries.
PARALLEL BOX PLOTS: SHOE SALES PER DAY
Store A: |-------[ | ]-------|
10 25 30 45 60
Store B: |------[ | ]-------|
35 50 55 70 85
<----+----+----+----+----+----+----+----+----+----+----+----+---->
0 10 20 30 40 50 60 70 80 90 100
From these parallel box plots, we can draw rigorous statistical conclusions:
1. Difference in Center: Store B has a much higher median daily sale (fifty-five pairs) than Store A (thirty pairs).
2. Comparison of Middle 50%: The IQR of Store A is forty-five minus twenty-five equals twenty pairs. The IQR of Store B is seventy minus fifty equals twenty pairs. Both stores exhibit identical middle variability.
3. Separation: The first quartile of Store B (fifty pairs) is greater than the third quartile of Store A (forty-five pairs). This demonstrates that over seventy-five percent of Store B's days exceed seventy-five percent of Store A's days.
Quantifying Separation as a Multiple of Variability
To express the difference between two population centers objectively, statisticians divide the difference in means by the measure of variability (MAD).
SEPARATION RATIO
Ratio = |Mean_1 - Mean_2| / MAD
(How many MADs apart are the two distribution centers?)
Suppose Group 1 has a mean height of sixty-two inches with a MAD of two inches, and Group 2 has a mean height of sixty-six inches with a MAD of two inches. Difference in means: sixty-six minus sixty-two equals four inches. Ratio: four inches divided by two inches equals two. The difference between the two group averages is exactly two times the Mean Absolute Deviation. A difference exceeding two MADs indicates substantial, statistically meaningful separation with minimal distributional overlap.
Exercises
Work through each problem carefully in your notebook before reading the provided solutions.
Exercise 1. Two soccer teams record the number of goals scored per match across twenty matches: Team Alpha: Mean = 3.2 goals, MAD = 0.8 goals Team Beta: Mean = 1.6 goals, MAD = 0.8 goals Calculate the difference in their mean goals, and express this difference as a multiple of the Mean Absolute Deviation.
Exercise 2. A parallel box plot compares the test scores of Class X and Class Y. Class X has a median of eighty-four and an IQR of eight. Class Y has a median of seventy-two and an IQR of ten. Compare the centers and variability of the two classes in complete sentences.
Exercise 3. A biologist compares the wingspans of two bird species. Species A has a mean wingspan of forty centimeters with a MAD of three centimeters. Species B has a mean wingspan of forty-two centimeters with a MAD of three centimeters. Is the difference between their means significant compared to their variability? Justify your answer using the separation ratio.
Exercise 4. Explain why it is inappropriate to use the mean to compare two distributions when one distribution is symmetric and the other distribution has severe extreme outliers.
Solutions
Solution to Exercise 1. Find the difference between the two means: three point two minus one point six equals one point six goals. Divide this difference by the common Mean Absolute Deviation of zero point eight: one point six divided by zero point eight equals two. The difference between the means of the two soccer teams is exactly two times the Mean Absolute Deviation, indicating a pronounced separation in scoring performance.
Solution to Exercise 2. In terms of central tendency, Class X performed substantially higher than Class Y, as demonstrated by Class X's median score of eighty-four compared to Class Y's median score of seventy-two (a difference of twelve points). In terms of variability, Class Y exhibited greater dispersion in its scores than Class X, reflected by Class Y's larger Interquartile Range of ten points compared to Class X's IQR of eight points. Class X was both higher-performing and more consistent.
Solution to Exercise 3. Calculate the difference between the two means: forty-two centimeters minus forty centimeters equals two centimeters. Compute the separation ratio by dividing the difference by the MAD of three centimeters: two divided by three equals zero point sixty-seven (two-thirds of a MAD). Because the difference in means is less than one full MAD (only 0.67 MADs), the difference is relatively small compared to the natural variability within each species. There is considerable overlap between the two wingspan distributions, indicating the difference is not highly pronounced.
Solution to Exercise 4. The mean is highly sensitive to extreme outliers, which pull the average artificially away from the true center of the data. If one distribution has severe outliers, its mean becomes skewed and distorted, making it an untruthful representation of typical values. Comparing a skewed mean to a symmetric mean compares two fundamentally different mathematical concepts. Instead, the median should be used to compare centers because the median is resistant to outliers, paired with the Interquartile Range (IQR) to compare variability.