Finding the location of a score in a distribution
Quantitative Specialists
Z score formula:
Where: X = a raw score (a value on some variable, X )
µ = the population mean σ = the population standard deviation
(Notice how both central tendency (µ) and variability (σ) are used to obtain a z score.)
Xz
Quantitative Specialists
Indicate the number of standard deviations a score is away from the mean.
Z-scores can be positive (above the mean), negative (below the mean), or zero (equal to the mean).
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Examples: z=+1; z=-2; z=0
z=+1; indicates that the score (X) is one standard deviation above the mean.
z=–2; indicates that a score (X) is two standard deviations below the mean.
z=0; indicates that a score (X) is zero standard deviations away from the mean (it is equal to the mean).
(Note: Non-integer z scores are possible as well, such a z = -.5.)
Quantitative Specialists
Example: X = 110, µ = 100, σ = 10
Interpretation: A score (X) of 110 is 1 standard deviation above the mean of 100.
110
10
10
100110
z
Xz
Quantitative Specialists
Example: X = 100, µ = 100, σ = 10
Interpretation: A score of 100 is 0 (zero) standard deviations away from the mean (it is equal to the mean).
010
0
10
100100
z
Xz
Quantitative Specialists
Example: X = 90, µ = 100, σ = 10
Interpretation: A score of 90 is one standard deviation below the mean.
110
10
10
10090
z
Xz
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Example: X = 120, µ = 100, σ = 10
Interpretation: A score of 120 is two standard deviations above the mean.
210
20
10
100120
z
Xz
Quantitative Specialists
Example: X = 95, µ = 100, σ = 10
Interpretation: A score of 95 is one-half (.5) of a standard deviation below the mean.
5.10
5
10
10095
z
Xz
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YouTube Channel: Quantitative Specialists
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1. X = 58, µ = 50, σ = 10
2. X = 74, µ = 65, σ = 6
3. X = 47, µ = 50, σ = 5
4. X = 87, µ = 100, σ = 8
5. X = 22, µ = 15, σ = 5
6. X = 77, µ = 70, σ = 4
7. X = 41, µ = 50, σ = 10
Xz
Quantitative Specialists
1. z = +.8
2. z = +1.5
3. X = -.6
4. X = -1.625
5. X = +1.4
6. X = +1.75
7. X = -.9
Xz
Quantitative Specialists
YouTube Channel: Quantitative Specialists
https://www.youtube.com/statisticsinstructor (typing “quantitative specialists” in the search box will also find us)
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