Standard Normal Distribution

Lecture 5

Dave Brocker

Farmingdale State College

Standard Normal Distribution

What do we know about normal distributions?

Standard Normal Distribution

Properties and Principles

The 4 properties of a standard normal distribution are:

• 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.

Standard Normal Distribution

Example 1

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

Where do most scores fall in relation to the mean?

• Most scores fall within one standard deviation of the mean.

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

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.

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

Where should we put the mean?

Standard Normal Distribution

Where should we put the mean?

Standard Normal Distribution

Mean: 6 | Standard Deviation: 1

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 the Netflix Original Dark a 7 or higher?

  • 15.8%

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 the Netflix Original Dark 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 the Netflix Original Dark 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.

Z-scores

Rule of Thumb

• 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 (AKA how much this particular participant is like the average person in the sample)

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

Calculating

Z-scores

• Subtract the mean from each X value.

• Divide by the standard deviation.

Tip

\[ z = \frac{x-\bar{x}}{s} \]

Z-scores

Z-Scores are calculated by

• Centering the X values on the mean: When we center the mean (AKA 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	x-xbar	(x-xbar)^2
10	-10	100
10	-10	100
20	0	0
30	10	100
30	10	100

Z-scores

Formula in Action

Tip

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

Tip

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

Z-scores

x	x-xbar	(x-xbar)^2	z
10	-10	100	-1
10	-10	100	-1
20	0	0	0
30	10	100	1
30	10	100	1

Z-scores

• 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.

• BUT: 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-Scores

Why Do We Care?

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

• 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.