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#ofSTDEVs is often called a " z -score"; we can use the symbol z . In symbols, the formulas become:

Sample x = x + zs z = x     x ¯ s
Population x = μ + z = x     μ σ

Two students, John and Ali, from different high schools, wanted to find out who had the highest GPA when compared to his school. Which student had the highest GPA when compared to his school?

Student GPA School Mean GPA School Standard Deviation
John 2.85 3.0 0.7
Ali 77 80 10

For each student, determine how many standard deviations (#ofSTDEVs) his GPA is away from the average, for his school. Pay careful attention to signs when comparing and interpreting the answer.

z = # of STDEVs = value  mean standard deviation = x + μ σ

For John, z = # o f S T D E V s = 2.85 3.0 0.7 = 0.21

For Ali, z = # o f S T D E V s = 77 80 10 = 0.3

John has the better GPA when compared to his school because his GPA is 0.21 standard deviations below his school's mean while Ali's GPA is 0.3 standard deviations below his school's mean.

John's z -score of –0.21 is higher than Ali's z -score of –0.3. For GPA, higher values are better, so we conclude that John has the better GPA when compared to his school.

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Two swimmers, Angie and Beth, from different teams, wanted to find out who had the fastest time for the 50 meter freestyle when compared to her team. Which swimmer had the fastest time when compared to her team?

Swimmer Time (seconds) Team Mean Time Team Standard Deviation
Angie 26.2 27.2 0.8
Beth 27.3 30.1 1.4

For Angie: z = 26 .2 – 27 .2 0 .8 = –1.25

For Beth: z = 27 .3 30 .1 1. 4 = –2

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The following lists give a few facts that provide a little more insight into what the standard deviation tells us about the distribution of the data.

    For any data set, no matter what the distribution of the data is:

  • At least 75% of the data is within two standard deviations of the mean.
  • At least 89% of the data is within three standard deviations of the mean.
  • At least 95% of the data is within 4.5 standard deviations of the mean.
  • This is known as Chebyshev's Rule.

    For data having a distribution that is bell-shaped and symmetric:

  • Approximately 68% of the data is within one standard deviation of the mean.
  • Approximately 95% of the data is within two standard deviations of the mean.
  • More than 99% of the data is within three standard deviations of the mean.
  • This is known as the Empirical Rule.
  • It is important to note that this rule only applies when the shape of the distribution of the data is bell-shaped and symmetric. We will learn more about this when studying the "Normal" or "Gaussian" probability distribution in later chapters.

References

Data from Microsoft Bookshelf.

King, Bill.“Graphically Speaking.” Institutional Research, Lake Tahoe Community College. Available online at http://www.ltcc.edu/web/about/institutional-research (accessed April 3, 2013).

Chapter review

The standard deviation can help you calculate the spread of data. There are different equations to use if are calculating the standard deviation of a sample or of a population.

  • The Standard Deviation allows us to compare individual data or classes to the data set mean numerically.
  • s = ( x x ¯ ) 2 n 1 or s = f ( x x ¯ ) 2 n 1 is the formula for calculating the standard deviation of a sample. To calculate the standard deviation of a population, we would use the population mean, μ , and the formula σ = ( x μ ) 2 N or σ = f ( x μ ) 2 N .

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Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ... Step 2: Find each score's deviation from the mean. ... Step 3: Square each deviation from the mean. ... Step 4: Find the sum of squares. ... Step 5: Divide the sum of squares by n – 1 or N.
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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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