Choose a representation
Locate the median using cumulative frequency. Compare standard deviation through distances from each mean, and combine means by recovering group totals.
Worked example
A data set has standard deviation 4. Each observation is increased by 6. What is the new standard deviation? Adding a common constant shifts both observations and their mean equally. Their deviations from the mean remain unchanged. The added constant 6 cancels from every deviation: (x+6)−(mean+6)=x−mean; standard deviation stays 4. The requested value is 4.
Contrast a wrong operation
In that example, 10 follows this error: Add the translation to the spread measure. The correct requested value is 4. Explain which quantity the mistaken method is actually calculating.
Connect a second form
A frequency table records value 4 four times, value 9 twice and value 12 once. What is the median? Seven observations place the median at ordered position 4. The first value occupies positions 1 through 4. Total count=4+2+1=7; the median is observation 4. Positions 1 through 4 all equal 4, so median=4. The requested value is 4.
Transfer and check
A data set has standard deviation 5. Each observation is increased by 7. What is the new standard deviation? Pause before checking. Answer: 5. Adding a common constant shifts both observations and their mean equally. Their deviations from the mean remain unchanged. The added constant 7 cancels from every deviation: (x+7)−(mean+7)=x−mean; standard deviation stays 5.
Boundaries of the method
A larger mean does not imply larger spread. Equal ranges do not force equal standard deviations. Averaging group means without weights can misrepresent unequal group sizes. This lesson samples a PSAT/NMSQT objective; its examples are original and do not claim official question status or full coverage.
A data set has standard deviation 4. Each observation is increased by 6. What is the new standard deviation?
- Adding a common constant shifts both observations and their mean equally.
- Their deviations from the mean remain unchanged.
- The added constant 6 cancels from every deviation: (x+6)−(mean+6)=x−mean; standard deviation stays 4.
- The requested value is 4.
Compare the other answer choices
Report the shift rather than the original spread.
Report variance rather than standard deviation.
Multiply the spread by the additive shift.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
Math specifications