To find the standardized score (z-score) for a piece of data:

  1. Find the deviation for the piece of data by subtracting the mean from the values (step 2 below)

  2. Divide the deviation by the standardized deviation (step 6 below)

  3. The result is the standardized score (z-score)

    Z-scores show how many standard deviations away from the mean a piece of data is.

An example using the data set {3, 5, 7, 8, 12, 13, 15, 16, 16, 18, 19, 24}
(This example includes the steps for finding a standard deviation).

  1. Find the mean of your data by adding all the data values and dividing by 12 (there are 12 pieces of data)

    3 + 5 + 7+ 8+ 12+ 13+ 15+ 16+ 16+ 18+ 19+ 24 = 156

    156/12 = 13

  2. Subtract the mean from each piece of data to find the deviations

    3 - 13 = -10
    5 - 13 = -8
    7 - 13 = -6
    8 - 13 = -5
    12 - 13 = -1
    13 - 13 = 0
    15 - 13 = 2
    16 - 13 = 3
    16 - 13 = 3
    18 - 13 = 5
    19 - 13 = 6
    24 - 13 = 11

  3. Square each of the deviations

    (-10)^2 = 100
    (-8)^2 = 64
    (-6)^2 = 36
    (-5)^2 = 25
    (-1)^2 = 1
    (0)^2 = 0
    (2)^2 = 4
    (3)^2 = 9
    (3)^2 = 9
    (5)^2 = 25
    (6)^2 = 36
    (11)^2 = 121

  4. Find the mean of the squared deviations (this is the variance)

    100 + 64 + 36 + 25 + 1 + 0 + 4 + 9 + 9 + 25 + 36 + 121 = 430

    430/12 = 35.833... = variance

  5. Find the square root of the mean of the squared deviations (this is the standard deviation)

    SQRT(35.833 ...) = 5.98609... = standard deviation

  6. To find the z scores for each piece of data, divide the deviations from step 2 by the standard deviation from step 5.

    -10/5.98609... = -1.671...
    -8/5.98609... = -1.336...
    -6/5.98609... = -1.002...
    -5/5.98609... = -.8353...
    -1/5.98609... = -.1671...
    0/5.98609... = 0
    2/5.98609... = .33411...
    3/5.98609... = .50116...
    3/5.98609... = .50116...
    5/5.98609... = .83527...
    6/5.98609... = 1.0023...
    11/5.98609... = 1.8376...

    -1.671, -1.336, -1.002, -.835, -.167, 0, .334, .501, .501, .835, 1.00, 1.838 are the z-scores corresponding to the 12 pieces of data (accurate to 3 decimal places).

To find the standardized score (z-score) for a piece of data with a calculator:

  1. Clear your previous lists (L1, L2, L3, etc.)

    1. STAT » EDIT » 4:ClrList » 2nd » 1 »» 2nd » 2 » , » 2nd » 3 » ENTER

  2. Enter all of the data into list one

    1. STAT » EDIT » 1:EDIT

  3. Find the mean of list one

    1. STAT » CALC » 1:1Var Stats » 2nd » 1 » ENTER

    2. The mean is the first number listed (X bar)

  4. Subtract the mean from each piece of data to find the deviations

    1. STAT » EDIT » 1:EDIT

    2. Move to the right and up so your cursor is on L2

    3. Write the formula "L1 - X bar"

      • ALPHA » + » 2nd » 1 » - » VARS » 5:STATISTICS » 2:Xbar »ALPHA» + » ENTER

  5. Divide the deviation by the standardized deviation

    1. Move to the right and up so your cursor is on L3

    2. Write the formula "L2/σ x "

      • ALPHA » + » 2nd » 2 » / » VARS » 5:STATISTICS » 4:σ x »ALPHA» + » ENTER

  6.  The result is the standardized score (z-score)

  7. Z-scores show how many standard deviations away from the mean a piece of data is.