Z-scores

Lecture 5

Dave Brocker

Farmingdale State College

Standard Normal Distribution

What do we know about normal distributions?

  • They are shaped like a bell (“bell curve”).

  • They are symmetric.

  • They are unimodal.

  • The mean = median = mode.

Standard Deviation

Standard deviation is in the scale of the variable (x).

  • A standard deviation of 1 means a distance of 1 on the scale used to measure the variable.
  • Jonas scores a 98 on the exam. The average grade on the exam was 97 with a standard deviation of 1.
    • Find Jonas’ score on the normal distribution.

Standard Deviation

Standard deviation is in the scale of the variable (x).

  • Jonas scores a 98 on the exam. The average grade on the exam was 97 with a standard deviation of 1.
    • Find Jonas’ score on the normal distribution.

Standard Normal Distribution

Percentages and Proportions

  • 68.3% of the data will fall within 1 SD of the mean.

  • 95.4% of the data will fall within 2 SD of the mean.

  • 99.7% of the data will fall within 3 SD of the mean.

  • “Within one standard deviation” means +1 as well as -1 standard deviation.

Standard Normal Distribution

Example 1

Students’ ratings of the Netflix Original Dark (range = 1- 10) form a normal distribution with m = 6 and s = 1.

  • What percentage of Students rate it a 7 or higher?

  • What percentage of Students rate it at least a 4?

  • What percentage of Students rate it an 8 or lower?

Standard Normal Distribution

Example 1:

Students’ ratings of the Netflix Original Dark form a normal distribution with m = 6 and s = 1.

  • What percentage of Students rate it a 7 or higher?

  • 15.8% (13.6 [1SD] + 2.1[2SD] + .1[3SD])

Standard Normal Distribution

Example 1:

Students’ ratings of the Netflix Original Dark form a normal distribution with m = 6 and s = 1.

  • What percentage of Students rate it at least a 4?

  • 2.2%

Standard Normal Distribution

Example 1:

Students’ ratings of the Netflix Original Dark form a normal distribution with m = 6 and s = 1.

  • What percentage of Students rate it an 8 or lower?

  • 97.6%

Z-scores

What does it do?

A z-score tells you, in standard deviation units how far the x-value is from the mean.

  • Z-scores are better than using raw SD

  • When the SD is a decimal, it is hard to find the exact point under the standard normal curve.

Z-scores

Rule of Thumb

  • The distance from the mean to the 1 on this standard normal curve is equal to the SD.
  • The distance from the mean to the 1 on this standard normal curve is equal to z=1.

Z-scores:

Find: Z = 1 | Z = -2 | Z = 0.5

Z-scores

What do they do!

  • Z-scores re-express the original data points (the x’s) in a way that intuitively lets us know:

  • How close the x is to the mean (how much this particular participant is like the average person in the sample)

  • Where it falls in the dispersion of the distribution (how different this particular participant is from the majority of people in the sample)

Calculating

Z-scores

Formula

\[ z = \large{\frac{\color{orange}{x}-\color{red}{\bar{x}}}{\color{pink}{s}}} \]

  • Subtract each x value from the mean.

  • Divide by the standard deviation

Practice Examples

Example 1

z = \(\frac{x - \mu}{\sigma}\)

  • Mean (\(\mu\)) = 50

  • Standard deviation (\(\sigma\)) = 10

  • Raw score (x) = 65

  • Find the z-score.

  • \(\frac{65-50}{10} = 1.5\)

Practice Examples

Example 2

  • Mean (\(\mu\)) = 100

  • Standard deviation (\(\sigma\)) = 15

  • Raw score (x) = 85

  • Find the z-score.

  • \(\frac{85-100}{15} = -1\)

Practice Examples

Example 3

  • Mean (\(\mu\)) = 70

  • Standard deviation (\(\sigma\)) = 8

  • Raw score (x) = 66

  • Find the z-score.

  • $ = .5

Practice Examples

Example 4

  • Mean (\(\mu\)) = 500

  • Standard deviation (\(\sigma\)) = 120

  • Raw score (x) = 740

  • Find the z-score.

  • \(\frac{740-500}{120} = 2\)

Practice Examples

Example 5

  • Mean (\(\mu\)) = 40

  • Standard deviation (\(\sigma\)) = 6

  • z-score = +2.5

  • Find the raw score (x).

  • \(2.5 = \frac{x-40}{6}\)

  • \(x = z\times s = 2.5\times6=15\)

Z-scores

Z-Scores are calculated by

  • Centering the X values on the mean: When we center the mean (mean-centering), we set the mean to 0.

  • Dividing by the standard deviation

  • When we divide by the SD, the space from the mean is expressed in standard deviations.

Z-scores

A Familiar Face: \(\bar{x} = 20\)

\(x\) \(\bar{x}\) \(x-\bar{x}\) \((x-\bar{x})^2\)
10  20  −10  100 
10  20  −10  100 
20  20    0    0 
30  20   10  100 
30  20   10  100 

Z-scores

Formula in Action

Average

\[\color{red}{\bar{x}} = \frac{\sum{x}}{n}\]

Standard Deviation

\[s^2 = \sqrt{\frac{\sum(x-\color{red}{\bar{x}})^2}{n-1}} = \frac{400}{4} = \sqrt{100} = s\]

Z-Score

\[z =\frac{x-\color{red}{\bar{x}}}{s} = \frac{20-10}{s}=\frac{-10}{s}\]

Z-scores

\(x\) \(x-\bar{x}\) \((x-\bar{x})^2\) \(z=\frac{(x-\bar{x})^2}{s}\)
10  −10  100  −1 
10  −10  100  −1 
20    0    0   0 
30   10  100   1 
30   10  100   1 
  • We made the mean = 0: When you mean-center a distribution, you shift it along the number line.

  • We made the SD = 1: When you divide a distribution by the SD, you shrink the distribution down.

  • The shape of the distribution remains the same.

Z-scores

  • Shift distribution along the number line.

  • Shrink distribution down.

  • The shape of the distribution remains the same.

Z-Scores

Why Do We Care?

  • A z-score tells me where my score falls in SD units.

  • I can then look at this standard normal curve, and estimate what percentage of people did better or worse than me.

Z-Scores

Why Do We Care?

The mean score for Exam 1 was a 92 with a standard deviation of 3.

  • Esmeralda scored an 86.

  • What percent of the class scored better than Esmeralda?

  • \(z=\frac{x-\bar{x}}{s} = \frac{86-92}{3} = -2\)

  • 98% of the class did better than Esmeralda.

Z-Scores

The mean score for Exam 1 was a 92 with a standard deviation of 3.

  • Jonas scored a 95.

  • What percent of the class scored better than Jonas?

  • \(z = \frac{95-92}{3} = 1\)

  • 16% scored higher than Jonas.

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