• Normal Distribution – A bell-shaped and symmetrical theoretical distribution, with the mean, the median, and the mode all coinciding at its peak and with frequencies gradually decreasing at both ends of the curve.

Normal Distributions

The Normal Distribution

Properties Of Normal Curve

• Normal curves are symmetrical.

• Normal curves are unimodal.

• Normal curves have a bell-shaped form.

• Mean, median, and mode all have the same value.

Normal Distribution

Many sets of data have common characteristics in how they are distributed. One of the most important probability distributions is the normal distribution. When a set of data forms a bell shape when plotted in a histogram, it is said to be normally distributed.

0

100

200

300

400

500

600

700

800

012345678

Example: The results of tossing 8 coins 2540 times were recorded and plotted:

Number of Tails

0 1 2 3 4 5 6 7 8

Fre

qu

enc

y

9.3.2

0

0.05

0.1

0.15

0.2

0.25

0.3

Number of Tails

0 1 2 3 4 5 6 7 8

Pro

bab

ilit

yNormal Distribution [cont’d]

You can convert the data in a histogram to a probability distribution:

9.3.3

Probability Distribution

The Relative Positions of the Mean, Median, and Mode: Symmetric Distribution

Mean

=Median

=Mode

M o d e

M ed ia n

M ea n

3- 7

Symmetric distributionSymmetric distribution: A distribution having the same shape on either side of the center

Skewed distributionSkewed distribution: One whose shapes on either side of the center differ; a nonsymmetrical distribution.

Can be positively or negatively skewed, or bimodal

3- 8

Shape of a Distribution

• Describes how data are distributed

• Measures of shape– Symmetric or skewed

Mean = Median Mean < Median Median < Mean

Right-SkewedLeft-Skewed Symmetric

ShapeShape

• Describes how data are distributed

• Measures of shape

Right-SkewedLeft-Skewed Symmetric

Mean = Median = ModeMean Median Mode Mode Median Mean

The Relative Positions of the Mean, Median and the Mode

More properties of normal curves

The 68-95-99.7 Rule

Standard deviation and the normal distribution

The Empirical Rule

The Empirical Rule

• μ ± 2σ covers about 95% of X’s

• μ ± 3σ covers about 99.7% of X’s

xμ

2σ 2σ

xμ

3σ 3σ

95.44% 99.72%

Percent of Values Within One Standard Deviations

68.26% of Cases

Percent of Values Within Two Standard Deviations

95.44% of Cases

Percent of Values Within Three Standard Deviations

99.72% of Cases

The Normal Distribution Curve

Basic Properties of the Normal Distribution Curve:

• The total area under the curve is 1.• It is symmetrical about the mean.• Approximately 68.3% of the data lies within 1 standard deviation of the mean.• Approximately 95.4% of the data lies within 2 standard deviations of the mean.• Approximately 99.7% of the data lies within 3 standard deviations of the mean. 9.3.5

Standard Scores

• One use of the normal curve is to explore Standard Scores. Standard Scores are expressed in standard deviation units, making it much easier to compare variables measured on different scales.

• There are many kinds of Standard Scores. The most common standard score is the ‘z’ scores.

• A ‘z’ score states the number of standard deviations by which the original score lies above or below the mean of a normal curve.

The Standard Normal Curve

• The Standard Normal Curve (z distribution) is the distribution of normally distributed standard scores with mean equal to zero and a standard deviation of one.

• A z score is nothing more than a figure, which represents how many standard deviation units a raw score is away from the mean.

An entire population of scores is transformed

into z-scores. The transformation does not change the shape of the population but the

mean is transformed into a value of 0 and the standard deviation is transformed to a value of

1.

Following a z-score transformation, the X-axis is relabled

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.

The Standardized Normal

• Any normal distribution (with any mean and standard deviation combination) can be transformed into the standardized normal distribution (Z)

• Need to transform X units into Z units

Z Scores Help in Comparisons

• One method to interpret the raw score is to transform it to a z score.

• The advantage of the z score transformation is that it takes into account both the mean value and the variability in a set of raw scores.

Translation to the Standardized Normal Distribution

• Translate from X to the standardized normal (the “Z” distribution) by subtracting the mean of X and dividing by its standard deviation:

σ

μXZ

Z always has mean = 0 and standard deviation = 1

The Z-score

The z-score is a conversion of the raw score into a standard score based on the mean and the standard deviation.

z-score = Raw score – Mean

Standard Deviation

Raw Score = 65

Standard Deviation = 15

65 – 55

15

= 0.67

Mean = 55

z-scoreExample

Standardize theNormal Distribution

X

X

Normal DistributionNormal Distribution

= 0

= 1

Z = 0

= 1

Z

ZX

ZX

Standardized

Normal DistributionStandardized

Normal Distribution

• The sample z-score is calculated by subtracting the sample mean from the individual raw score and then dividing by the sample standard deviation.

Where : x is the individual score x is the sample mean

s is the sample standard deviation.

z = x - x s

• The population z-score is calculated by subtracting the population mean from the individual raw score and then dividing by the population standard deviation.

Where : x is the individual score µ is the population mean

is the population standard deviation

Z = X - µ

Z-Scores and the Normal Distribution

• If we have a normal distribution we can make the following assumptions.

• Approximately 68% of the scores are between a z-score of 1 and -1.

• Approximately 95% of the scores will be between a z-score of 2 and -2.

• Approximately 99.7% of the scores will be between a z-score of 3 and -3.

Total Area = 1; This represents 100% ofThe data set.

mean x

0.5 0.5

Represent those scoresabove the mean, i.e., 50% of the data set.

Represents those scoresbelow the mean, i.e., 50% of the data set.

• The z-score for x gives the area from x to the mean. This represents the percentage of those in the data set that score between x and the mean. To get percentile for x, we add this to 0.5 from the first part of the distribution

Example

• If X is distributed normally with mean, , of 100 and standard deviation, , of 50, the Z value for X = 200 is

• This says that X = 200 is two standard deviations above the mean of 100.

2.050

100200

σ

μXZ

35

Application of 68-95-99.7 rule• Male height has a Normal distribution with μ = 70.0

inches and σ = 2.8 inches

• Notation: Let X ≡ male height; X~ N(μ = 70, σ = 2.8)

68-95-99.7 rule

• 68% in µ = 70.0 2.8 = 67.2 to 72.8

• 95% in µ 2 = 70.0 2(2.8) = 64.4 to 75.6

• 99.7% in µ 3 = 70.0 3(2.8) = 61.6 to 78.4

36

Application: 68-95-99.7 RuleWhat proportion of men are less than 72.8 inches tall?μ + σ = 70 + 2.8 = 72.8 (i.e., 72.8 is one σ above μ)

?

70 72.8 (height) +1

84%

68% (by 68-95-99.7 Rule)

16%

-1

Therefore, 84% of men are less than 72.8” tall.

16%68% (total AUC = 100%)

37

Finding Normal proportionsWhat proportion of men are less than 68” tall? This is equal to the AUC to the left of 68 on X~N(70,2.8)

?

68 70 (height values)

To answer this question, first determine the z-score for a value of 68 from X~N(70,2.8)

38

Z score

• The z-score tells you how many standard deviation the value falls below (negative z score) or above (positive z score) mean μ

• The z-score of 68 when X~N(70,2.8) is:

71.08.2

7068

x

z

zx

Thus, 68 is 0.71 standard deviations below μ.

39

Example: z score and associate value

-0.71 0 (z values)68 70 (height values)

?

04/20/23 Chapter 3 40

Normal Cumulative Proportions (Table A)

z .00 .02

0.8 .2119 .2090 .2061

.2420 .2358

0.6 .2743 .2709 .2676

0.7

.01

.2389

Thus, a z score of −0.71 has a cumulative proportion of .2389

41

Area to the right (“greater than”)

.2389

-0.71 0 (z values)68 70 (height values)

1.2389 = .7611

Since the total AUC = 1:AUC to the right = 1 – AUC to leftExample: What % of men are greater than 68” tall?

Finding the Area Under the Curve

1. Find the area between z-scores -1.22 and 1.44.

-1.22 1.44

-1.22 1.44

The area for z-score -1.22 is 0.1112.

The area for z-score 1.44 is 0.9251.Therefore, the area between z-scores -1.22 and 1.44 is

0.9251 - 0.1112 = 0.8139.

Finding the Area Under the Curve2. Find the area between the mean and z-score -1.78.

-1.78

The area for z-score -1.78 is 0.0375.

Therefore, the area between the mean and z-score -1.78 is0.5 - 0.0375 = 0.4625.

3. Find the area between the mean and z-score 1.78.

1.78

The area for z-score 1.78 is 0.9625.Therefore, the area between the mean and z-score 1.78 is

0.9625 - 0.5 = 0.4625.

Finding the Area Under the Curve

4. Find the area greater than z-score -0.68.

5. Find the area greater than z-score 1.40.

-0.68

1.40

The area for z-score -0.68is 0.2483.Therefore, the area betweenthe mean and z-score -0.68 is1 - 0.2483 = 0.7515.

The area for z-score 1.40 is 0.9192.Therefore, the area between the mean and z-score 1.40 is1 - 0.9192 = 0.0808.