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Z-Scores worksheets

Calculate z-scores, recover original values, and compare relative positions across data sets with positive standard deviations.

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Use z = (x - mean) / standard deviation. The given standard deviation is positive. Explain whether each value is above or below its mean; no normal-distribution assumption is needed.

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Keep the convention visible

Variance worksheets state whether each data set is a population or sample and show the corresponding denominator. Z-score worksheets provide the mean and a positive standard deviation, so students can focus on centered, scaled comparisons.

Learning objective

What this activity builds

Use z = (x - mean) / standard deviation to describe and compare a value's position relative to its own data set.

Worked example

A value is 74 in a data set with mean 68 and standard deviation 3. Find its z-score.

z = (74 - 68) / 3 = 2.

The value is two standard deviations above its mean. This interpretation does not require the data to be normally distributed.

Teaching notes

Make the practice count

  • Read the sign before interpreting the size of a z-score.
  • When comparing data sets, standardize each value with its own mean and standard deviation.
  • Use context before describing a larger standardized position as better or worse.

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Math SheetNameScoreTeacherDateZ-Scores and Standardized ValuesUse z = (x - mean) / standard deviation. The given standard deviation is positive. Explain whether each value is above or below its mean; no normal-distribution assumption isneeded.1.Data set A: x = 52.75, mean = 48, SD = 9.5 points. Data set B: x = 23, mean = 36, SD = 6.5 centimeters. Which value has the greater standardizedposition?SetxmeanSDunitA52.75489.5pointsB23366.5centimeters2.Data set A measurement has mean 89 points and standard deviation 11 points. Find x when z = 0.5.SetxmeanSDunitA?8911pointsState the convention and include units in the final answer.math-sheet.comPage 1 of 3
Math SheetNameScoreTeacherDateZ-Scores and Standardized ValuesUse z = (x - mean) / standard deviation. The given standard deviation is positive. Explain whether each value is above or below its mean; no normal-distribution assumption isneeded.3.Data set A measurement has mean 86.5 minutes and standard deviation 9.5 minutes. Find the z-score for 81.75 minutes.SetxmeanSDunitA81.7586.59.5minutes4.Data set A: x = 71, mean = 57.5, SD = 13.5 degrees. Data set B: x = 29.75, mean = 45.5, SD = 10.5 minutes. Which value has the greaterstandardized position?SetxmeanSDunitA7157.513.5degreesB29.7545.510.5minutesState the convention and include units in the final answer.math-sheet.comPage 2 of 3
Math SheetNameScoreTeacherDateZ-Scores and Standardized ValuesUse z = (x - mean) / standard deviation. The given standard deviation is positive. Explain whether each value is above or below its mean; no normal-distribution assumption isneeded.5.Data set A measurement has mean 51 points and standard deviation 10 points. Find x when z = 0.5.SetxmeanSDunitA?5110points6.Data set A measurement has mean 29 points and standard deviation 2.5 points. Find the z-score for 30.25 points.SetxmeanSDunitA30.25292.5pointsState the convention and include units in the final answer.math-sheet.comPage 3 of 3

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