Linear Transformations of Data: Effects on Statistics
Linear transformations of data in Algebra 1 (California Reveal Math, Grade 9) show how summary statistics change when every value in a data set is shifted or scaled. Adding a constant c shifts the mean, median, and quartiles by c, but does NOT change standard deviation, IQR, or range. Multiplying by a constant k scales the mean, median, standard deviation, IQR, and range all by k. Understanding these rules helps students predict the effect of unit changes (like converting Celsius to Fahrenheit) on statistical summaries without recalculating from scratch.
Key Concepts
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).
Common Questions
What happens to the mean when you add a constant to all data values?
The mean increases by the same constant. For example, if every score increases by 5, the mean increases by 5.
What happens to the standard deviation when you add a constant to all values?
The standard deviation does NOT change. Adding a constant shifts all values equally, so their spread relative to the mean is unchanged.
What happens when you multiply all data values by a constant?
The mean, median, quartiles, standard deviation, IQR, and range all multiply by the same constant. Multiplying by k scales every statistic by k.
Why doesn't adding a constant change the standard deviation?
Standard deviation measures spread — the distances between values. Adding a constant moves all values the same amount, preserving all relative distances.
Where are linear transformations of data covered in California Reveal Math Algebra 1?
This topic is taught in California Reveal Math, Algebra 1, as part of Grade 9 statistics and data analysis.
What real-world example uses this concept?
Converting temperatures from Celsius to Fahrenheit (F = 1.8C + 32) is a linear transformation: the 1.8 factor scales spread; the +32 shifts measures of center but not spread.
What statistics are NOT affected by adding a constant?
Standard deviation, IQR, and range are unaffected by adding a constant because they measure spread (differences between values), which does not change when all values shift equally.