Z scores
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This document discusses z-scores and how they are used to standardize distributions. A z-score specifies the position of a data point within a distribution by measuring its distance from the mean in units of standard deviations. The mean of a standardized distribution with z-scores is 0 and the standard deviation is 1. Converting raw scores to z-scores transforms distributions to have the same shape while accounting for different means and standard deviations. Z-scores allow for comparison of data points from different distributions.
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Z scores
In statistics, the standard score is the (signed) number of standard deviations an observation or datum is above the mean. Thus, a positive standard score represents a datum above the mean, while a negative standard score represents a datum below the mean. It is a dimensionless quantity obtained by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. This conversion process is called standardizing or normalizing (however, "normalizing" can refer to many types of ratios; see normalization (statistics) for more).Standard scores are also called z-values, z-scores, normal scores, and standardized variables; the use of "Z" is because the normal distribution is also known as the "Z distribution". They are most frequently used to compare a sample to a standard normal deviate (standard normal distribution, with μ = 0 and σ = 1), though they can be defined without assumptions of normality.
Z scores
The document discusses the importance of converting raw scores to standardized z-scores. It provides the formula for calculating a z-score and explains that a z-score indicates how many standard deviations a score is from the mean. Several examples are given of calculating z-scores based on different hypothetical scores and population means and standard deviations. The key benefits of z-scores are that they allow for comparison of scores from different distributions and indicate whether a score is above, below, or at the average.
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The document discusses z-scores and how they are used to standardize scores in a normal distribution. A z-score represents the number of standard deviations a data point is from the mean. It is calculated by subtracting the mean from the data value and dividing by the standard deviation. Examples are given to demonstrate how to calculate z-scores and use them to determine percentiles and the percentage of cases above or below a given score.
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In statistics, the standard score is the (signed) number of standard deviations an observation or datum is above the mean. Thus, a positive standard score represents a datum above the mean, while a negative standard score represents a datum below the mean. It is a dimensionless quantity obtained by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. This conversion process is called standardizing or normalizing (however, "normalizing" can refer to many types of ratios; see normalization (statistics) for more).Standard scores are also called z-values, z-scores, normal scores, and standardized variables; the use of "Z" is because the normal distribution is also known as the "Z distribution". They are most frequently used to compare a sample to a standard normal deviate (standard normal distribution, with μ = 0 and σ = 1), though they can be defined without assumptions of normality.
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The document discusses the importance of converting raw scores to standardized z-scores. It provides the formula for calculating a z-score and explains that a z-score indicates how many standard deviations a score is from the mean. Several examples are given of calculating z-scores based on different hypothetical scores and population means and standard deviations. The key benefits of z-scores are that they allow for comparison of scores from different distributions and indicate whether a score is above, below, or at the average.
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The document discusses z-scores and how they are used to standardize scores in a normal distribution. A z-score represents the number of standard deviations a data point is from the mean. It is calculated by subtracting the mean from the data value and dividing by the standard deviation. Examples are given to demonstrate how to calculate z-scores and use them to determine percentiles and the percentage of cases above or below a given score.
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The document discusses properties of normal distributions and the standard normal distribution. It provides examples of finding probabilities and values associated with normal distributions. The key points are:
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This document discusses statistical concepts like inferential statistics, normal distributions, z-scores, t-scores, standardization, and correlations. Some key points covered include:
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- 2. Normal curve distribution comparisons 12 82 x = 76 (a) X = 76 is slightly below average 12 70 x = 76 (b) X = 76 is slightly above average 3 70 x = 76 (c) X = 76 is far above average
- 4. Normal curve segmented by sd’s X z -2 -1 0 +1 +2
- 6. Z formula
- 7. Normal curve with sd’s = 4 60 64 68 µ X 66
- 9. To get the raw score from the z-score:
- 11. Z-distribution X z -2 -1 0 +1 +2 100 110 120 90 80
- 12. A small population 0, 6, 5, 2, 3, 2 N=6 x x - µ (x - µ) 2 0 0 - 3 = -3 9 6 6 - 3 = +3 9 5 5 - 3 = +2 4 2 2 - 3 = -1 1 3 3 - 3 = 0 0 2 2 - 3 = -1 1
- 13. Frequency distributions with sd’s 2 1 0 1 2 3 4 5 6 µ X frequency (a) 2 1 -1.5 -1.0 -0.5 0 +0.5 +1.0 +1.5 µ z frequency (b)
- 14. Let’s transform every raw score into a z-score using: x = 0 x = 6 x = 5 x = 2 x = 3 x = 2 = -1.5 = +1.5 = +1.0 = -0.5 = 0 = -0.5
- 15. Z problem computation Mean of z-score distribution : Standard deviation: z z - µ z (z - µ z ) 2 -1.5 -1.5 - 0 = -1.5 2.25 +1.5 +1.5 - 0 = +1.5 2.25 +1.0 +1.0 - 0 = +1.0 1.00 -0.5 -0.5 - 0 = -0.5 0.25 0 0 - 0 = 0 0 -0.5 -0.5 - 0 = -0.5 0.25
- 16. Psychology vs. Biology 10 50 µ X = 60 X Psychology 4 48 µ X = 56 X Biology 52
- 17. Converting Distributions of Scores
- 18. Original vs. standardized distribution 14 57 µ Joe X = 64 z = +0.50 X Original Distribution 10 50 µ X Standardized Distribution Maria X = 43 z = -1.00 Maria X = 40 z = -1.00 Joe X = 55 z = +0.50
- 19. Correlating Two Distributions of Scores with z-scores
- 20. Distributions of height and weight = 4 µ = 68 Person A Height = 72 inches Distribution of adult heights (in inches) = 16 µ = 140 Person B Weight = 156 pounds Distribution of adult weights (in pounds)
Editor's Notes
- Figure 5.6 Following a z -score transformation, the X -axis is relabeled in z -score units. The distance that is equivalent to 1 standard deviation on the X -axis (σ = 10 points in this example) corresponds to 1 point on the z -score scale.