Learn on PengiCalifornia Reveal Math, Algebra 1Unit 11: Statistics

11-4 Comparing Sets of Data

In this Grade 9 lesson from California Reveal Math Algebra 1, Unit 11, students learn how to compare two or more data sets by selecting appropriate measures of center and spread based on distribution shape. For symmetric distributions, students use mean and standard deviation, while for skewed distributions they apply the five-number summary and interquartile range. Students also use graphing technology to construct histograms and box plots to analyze and interpret real-world data.

Section 1

Constructing and Comparing Double Displays

Property

To visually compare two different data sets, you must construct their graphs side-by-side or stacked, using identical axis scales.

Double Histograms: When comparing two histograms, you must use the exact same bin width ( $w$ ) and the same numerical scale on both the horizontal (x-axis) and vertical (y-axis) axes.
Double Box Plots: Draw both box-and-whisker plots floating parallel to each other over the exact same horizontal number line.

Examples

The Scale Trap (Histograms): Class A scores range from 50–100. Class B ranges from 55–98. Both histograms MUST use the same bins (e.g., width of 10 starting at 50). If you use a bin width of 10 for Class A and 5 for Class B, Class B's data will falsely appear much more spread out than it actually is.
The Scale Trap (Vertical Axis): Class A and B both have 30 students. If Class A's tallest bar reaches 12, both y-axes must be scaled to at least 12. If Class B's axis stops at 6, its bars will physically look twice as tall as Class A's, creating a misleading visual comparison.
Comparing Double Box Plots: Dataset X has a median of 0.45 seconds and an IQR of 0.10. Dataset Y has a median of 0.52 seconds and an IQR of 0.25. When drawn on the same number line, you can instantly see that Group X not only has a faster (lower) center but is also much more consistent (shorter box).

Explanation

Section 2

Using Technology to Compute and Compare

Property

When comparing multiple large data sets, calculating statistics by hand is inefficient and prone to error. Spreadsheets and graphing calculators (like the TI-84) automate these calculations and can instantly generate comparative graphs.

Spreadsheet Functions: =AVERAGE(range) finds the mean, =MEDIAN(range) finds the center, =STDEV(range) finds standard deviation, and =QUARTILE(range, 1 or 3) finds the edges of the box plot.
Graphing Calculators: Entering data into two lists (L1 and L2) and running 1-Var Stats returns the entire 5-Number Summary, Mean ( $\bar{x}$ ), and Standard Deviation ( $s$ ) simultaneously.

Examples

Spreadsheet Efficiency: A student enters Morning quiz scores in Column A and Afternoon scores in Column B. By typing =STDEV(A1:A20) and =STDEV(B1:B20), she instantly sees that the Afternoon class has a standard deviation of 11.8 compared to the Morning's 6.2, proving the Afternoon class was much less consistent.
TI-84 Comparison: A class enters two data sets into L1 and L2. The calculator yields $\bar{x}_1 = 78.4$ , $\text{Med}_1 = 79$ (Symmetric) and $\bar{x}_2 = 71.0$ , $\text{Med}_2 = 68$ (Skewed Right). The student can then use the STAT PLOT menu to stack two box plots on the screen to visually confirm the skewness.

Explanation

Technology is your best friend when comparing data. It instantly computes the heavy math, freeing up your brain to actually analyze what the numbers mean. However, technology is blind—it will happily calculate the Mean and Standard Deviation for a massively skewed data set. It is always up to you, the human, to look at the graphs generated by the software and decide whether those statistics actually represent the truth.

Section 3

Linear Transformations of Data: Effects on Statistics

Property

When you mathematically adjust every single value in a data set by adding a constant or multiplying by a constant, the summary statistics change according to strict algebraic rules. Let the new data be $x_{\text{new}} = m \cdot x + k$ .

Adding/Subtracting a Constant ( $+ k$ ): Shifts the Center (Mean, Median) by $k$ . Does NOT change the Spread (Range, IQR, Standard Deviation).
Multiplying/Dividing by a Constant ( $\times m$ ): Scales the Center (Mean, Median) by $m$ . Scales the Spread (Range, IQR, Standard Deviation) by $m$ .

Examples

Adding a Constant: A data set of test scores has a Mean = 50 and Standard Deviation = 6. The teacher decides to curve the test by giving everyone +5 bonus points.

The new Mean shifts up to $50 + 5 = 55$ .
The new Standard Deviation stays exactly $6$ (because the students are still spread out by the exact same amount; everyone just moved up the number line together).

Multiplying by a Constant: A data set of measurements in feet has a Median = 20 and an IQR = 8. You want to convert the data to inches, so you multiply every value by 12.

The new Median is $20 \times 12 = 240$ inches.
The new IQR is $8 \times 12 = 96$ inches. Everything stretches.

Combined Transformation: Test scores have a Mean = 70 and Standard Deviation = 8. A teacher curves grades by multiplying by $1.1$ and then adding $5$ .

New Mean: $1.1(70) + 5 = 82$ .
New Standard Deviation: $1.1(8) = 8.8$ (the $+5$ does not affect the spread).

Explanation

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Constructing and Comparing Double Displays

Property

To visually compare two different data sets, you must construct their graphs side-by-side or stacked, using identical axis scales.

Double Histograms: When comparing two histograms, you must use the exact same bin width ( $w$ ) and the same numerical scale on both the horizontal (x-axis) and vertical (y-axis) axes.
Double Box Plots: Draw both box-and-whisker plots floating parallel to each other over the exact same horizontal number line.

Examples

The Scale Trap (Histograms): Class A scores range from 50–100. Class B ranges from 55–98. Both histograms MUST use the same bins (e.g., width of 10 starting at 50). If you use a bin width of 10 for Class A and 5 for Class B, Class B's data will falsely appear much more spread out than it actually is.
The Scale Trap (Vertical Axis): Class A and B both have 30 students. If Class A's tallest bar reaches 12, both y-axes must be scaled to at least 12. If Class B's axis stops at 6, its bars will physically look twice as tall as Class A's, creating a misleading visual comparison.
Comparing Double Box Plots: Dataset X has a median of 0.45 seconds and an IQR of 0.10. Dataset Y has a median of 0.52 seconds and an IQR of 0.25. When drawn on the same number line, you can instantly see that Group X not only has a faster (lower) center but is also much more consistent (shorter box).

Explanation

Section 2

Using Technology to Compute and Compare

Property

Spreadsheet Functions: =AVERAGE(range) finds the mean, =MEDIAN(range) finds the center, =STDEV(range) finds standard deviation, and =QUARTILE(range, 1 or 3) finds the edges of the box plot.
Graphing Calculators: Entering data into two lists (L1 and L2) and running 1-Var Stats returns the entire 5-Number Summary, Mean ( $\bar{x}$ ), and Standard Deviation ( $s$ ) simultaneously.

Examples

Spreadsheet Efficiency: A student enters Morning quiz scores in Column A and Afternoon scores in Column B. By typing =STDEV(A1:A20) and =STDEV(B1:B20), she instantly sees that the Afternoon class has a standard deviation of 11.8 compared to the Morning's 6.2, proving the Afternoon class was much less consistent.
TI-84 Comparison: A class enters two data sets into L1 and L2. The calculator yields $\bar{x}_1 = 78.4$ , $\text{Med}_1 = 79$ (Symmetric) and $\bar{x}_2 = 71.0$ , $\text{Med}_2 = 68$ (Skewed Right). The student can then use the STAT PLOT menu to stack two box plots on the screen to visually confirm the skewness.

Explanation

Section 3

Linear Transformations of Data: Effects on Statistics

Property

Adding/Subtracting a Constant ( $+ k$ ): Shifts the Center (Mean, Median) by $k$ . Does NOT change the Spread (Range, IQR, Standard Deviation).
Multiplying/Dividing by a Constant ( $\times m$ ): Scales the Center (Mean, Median) by $m$ . Scales the Spread (Range, IQR, Standard Deviation) by $m$ .

Examples

Adding a Constant: A data set of test scores has a Mean = 50 and Standard Deviation = 6. The teacher decides to curve the test by giving everyone +5 bonus points.

Multiplying by a Constant: A data set of measurements in feet has a Median = 20 and an IQR = 8. You want to convert the data to inches, so you multiply every value by 12.

The new Median is $20 \times 12 = 240$ inches.
The new IQR is $8 \times 12 = 96$ inches. Everything stretches.

Combined Transformation: Test scores have a Mean = 70 and Standard Deviation = 8. A teacher curves grades by multiplying by $1.1$ and then adding $5$ .

New Mean: $1.1(70) + 5 = 82$ .
New Standard Deviation: $1.1(8) = 8.8$ (the $+5$ does not affect the spread).

Explanation

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Constructing and Comparing Double Displays

Property

To visually compare two different data sets, you must construct their graphs side-by-side or stacked, using identical axis scales.

Double Histograms: When comparing two histograms, you must use the exact same bin width ( $w$ ) and the same numerical scale on both the horizontal (x-axis) and vertical (y-axis) axes.
Double Box Plots: Draw both box-and-whisker plots floating parallel to each other over the exact same horizontal number line.

Examples

The Scale Trap (Histograms): Class A scores range from 50–100. Class B ranges from 55–98. Both histograms MUST use the same bins (e.g., width of 10 starting at 50). If you use a bin width of 10 for Class A and 5 for Class B, Class B's data will falsely appear much more spread out than it actually is.
The Scale Trap (Vertical Axis): Class A and B both have 30 students. If Class A's tallest bar reaches 12, both y-axes must be scaled to at least 12. If Class B's axis stops at 6, its bars will physically look twice as tall as Class A's, creating a misleading visual comparison.
Comparing Double Box Plots: Dataset X has a median of 0.45 seconds and an IQR of 0.10. Dataset Y has a median of 0.52 seconds and an IQR of 0.25. When drawn on the same number line, you can instantly see that Group X not only has a faster (lower) center but is also much more consistent (shorter box).

Explanation

Section 2

Using Technology to Compute and Compare

Property

Spreadsheet Functions: =AVERAGE(range) finds the mean, =MEDIAN(range) finds the center, =STDEV(range) finds standard deviation, and =QUARTILE(range, 1 or 3) finds the edges of the box plot.
Graphing Calculators: Entering data into two lists (L1 and L2) and running 1-Var Stats returns the entire 5-Number Summary, Mean ( $\bar{x}$ ), and Standard Deviation ( $s$ ) simultaneously.

Examples

Spreadsheet Efficiency: A student enters Morning quiz scores in Column A and Afternoon scores in Column B. By typing =STDEV(A1:A20) and =STDEV(B1:B20), she instantly sees that the Afternoon class has a standard deviation of 11.8 compared to the Morning's 6.2, proving the Afternoon class was much less consistent.
TI-84 Comparison: A class enters two data sets into L1 and L2. The calculator yields $\bar{x}_1 = 78.4$ , $\text{Med}_1 = 79$ (Symmetric) and $\bar{x}_2 = 71.0$ , $\text{Med}_2 = 68$ (Skewed Right). The student can then use the STAT PLOT menu to stack two box plots on the screen to visually confirm the skewness.

Explanation

Section 3

Linear Transformations of Data: Effects on Statistics

Property

Adding/Subtracting a Constant ( $+ k$ ): Shifts the Center (Mean, Median) by $k$ . Does NOT change the Spread (Range, IQR, Standard Deviation).
Multiplying/Dividing by a Constant ( $\times m$ ): Scales the Center (Mean, Median) by $m$ . Scales the Spread (Range, IQR, Standard Deviation) by $m$ .

Examples

Adding a Constant: A data set of test scores has a Mean = 50 and Standard Deviation = 6. The teacher decides to curve the test by giving everyone +5 bonus points.

Multiplying by a Constant: A data set of measurements in feet has a Median = 20 and an IQR = 8. You want to convert the data to inches, so you multiply every value by 12.

The new Median is $20 \times 12 = 240$ inches.
The new IQR is $8 \times 12 = 96$ inches. Everything stretches.

Combined Transformation: Test scores have a Mean = 70 and Standard Deviation = 8. A teacher curves grades by multiplying by $1.1$ and then adding $5$ .

New Mean: $1.1(70) + 5 = 82$ .
New Standard Deviation: $1.1(8) = 8.8$ (the $+5$ does not affect the spread).