**PROBABILITY &**

THE NORMAL CURVE

THE NORMAL CURVE

Other probability options

Other probability options

**Z-scores**

Finding a z-score continued

**Finding a z-score**

**Z-scores**

PROBABILITY

Other probability options

σ

USING TABLE A

1.25 σ

Using Table A - continued

Areas under the normal curve

μ = 100

σ = 10

A distribution of female IQ scores

A distribution of male IQ scores

Examples of normal curve

DEFINITION: The relative likelihood of occurrence of any given outcome or event

Always varies between 1.0 and 0

0.5 = 5 chances out of 10

0 = impossible (close to 0 = event is very unlikely)

1 = certainty (close to 1 = event is very likely)

FORMULA:

# of times the outcome or event can occur

total # of times any outcome or event can occur

P =

EXAMPLE: What is the probability of picking the ace of spades in a full deck of cards?

1/4

2/4

1/4

P = 1/52

P = 0.019

DISPLAYING PROBABILITY

The Probability Line

Displaying probability

Can be displayed like frequency distributions (i.e. in histograms, etc.)

DISPLAYING PROBABILITY

Like frequency distributions, probability distributions can take a variety of shapes

Probability continued

We can calculate the Mean of a probability distribution

Tells us the “most expected value” in the long run

Example: The mean number of heads in 2 coin flips is 1 because the “most expected” outcome is to get 1 head and 1 tail.

Symbols:

Mean in a probability distribution =

μ (mu)

Variance in a probability distribution =

σ 2 (sigma squared)

Standard Deviation in a probability distribution =

σ (sigma)

**the normal curve**

-Theoretical/ideal model

-Useful for:

Describing distributions

Interpreting standard deviation

Making statements about probability

Statistical decision-making

-Characteristics:

Symmetrical curve

Unimodal

What Is Normal?

Quite a lot!

Height

Weight

IQ Test scores

For many things, the bigger the sample, the more normal the distribution

What isn’t?

Income

Age

1, 2, 3, rule

68.26% of area between 1 and -1

95.44% of area between 2 and -2

99.74% of area between 3 and -3

PRACTICE:

A)

34.13% of IQ scores fall between 100 and what score (below the mean)?

For men?

For women?

B)

What is the bracket of IQ scores for 99.74% of women?

C)

What is the bracket of IQ scores for 68.26% of men?

μ = 100

σ = 15

Now we know about areas under the normal curve

(mean, standard deviation, percentages of cases that lie within units of standard deviation…)

**BUT WAIT!**

How do we determine the percentage of cases that lie between units of standard deviation?

FOR EXAMPLE -- what percentage of cases fall between the mean and 1.25 standard deviations above the mean?

That's where TABLE A comes in.

In this case:

z = 1.25

Area between mean and z = 39.44%

Area beyond z = 10.56%

Table A (pages 361-364) shows us:

(a) “z” -- stands for units of standard deviation

(b) “Area between mean and z” – in percentages (for only one side of the curve)

(c) “Area beyond z” – in percentages (for only one side of the curve)

Have you heard of these guys?

AKA “units of standard deviation”

AKA “sigma distances”

AKA “standard score”

How do we determine the z-score for a result within our data set?

EXAMPLE: Catherine Humes collected data about the number of jelly beans people ate in one sitting and found that:

the mean # was 13

the standard deviation was 2.

How can I find the z-score for 16 jelly beans?

μ = 13

σ = 2

1. Find the deviation from the mean

16 – 13 = 3

How?

***16 is 3 units above the mean***

2. Divide this total by the standard deviation

3 / 2 = 1.5

What does this mean? Give me the numbers.

***This tells us that 16 jelly beans has a z-score of 1.5***

***AKA 16 jelly beans is 1.5 standard deviations above the mean***

TO SUM UP:

-z-scores: direction and degree that any given score deviates from the mean. (in units of standard deviation)

-z-score of +1.4 indicates that a score lies 1.4σ units above the mean

-z-score of -2.1 indicates that a score lies 2.1σ units below the mean

FORMULA:

When we combine our understanding of the normal curve and z-scores (Table A)…

We can determine the probability of obtaining a score in a distribution

34.13% of scores fall between the mean and +1 σ

Convert the percentage to probability

P = 0.3413

Probability and the normal curve

Probability of getting a score between mean and +1 σ = 34.13 in 100

Finding the probability of scores on both sides of the curve

Example: Normalcurve, inc. (Founder loves normal curves and salaries are allocated and spread according to the normal curve)

What is the probability of getting a salary between $18,000 and $22,000 where the mean is $20,000 and the standard deviation is $1,500?

1) Find z-score for both options

P = 0.8164

Finding the probability of obtaining more than a single portion of the area under the curve

Example: At Normalcurve, inc., what is the probability of getting a salary either below $18,000 or above $22,000?

1) Find the z-score for both options

2) Find percentage (beyond the z) for both options using Table A

3) Convert percentage to probability for both options

4) Add probability for both options together

P = 0.1836

Finding the score value based on knowing score values for portions of the area under the normal curve

Example: At Normalcurve, inc., what is the salary level that defines the top 10.03% of earners?

1) Locate 10.03% in Table A (column c)

2) Locate z-score for 10.03% within Table A

3) Reverse our z-score formula

score – mean / standard deviation = z-score

z-score x standard deviation + mean = score

Part 1: Men’s Heights

The distribution of heights of 21 year-old men has a mean of 175 centimetres and a standard deviation of 6 cm. Using this information, answer the following questions and show your work.

1. What percentage of men are taller than 190 cm?

2. Between what heights do the middle 68.26% of men fall?

3. What is the probability of a man being shorter than 160 cm?

Part 2: Length of Pregnancies

The length of human pregnancies from conception to birth varies with a mean of 266 days and a standard deviation of 16 days. Using this information, answer the following questions and show your work.

1. Between what values do the middle 95% of all pregnancies fall?

2. How long are the longest 2.5% of pregnancies?

10% of earners = $ 21 920 or more

2) Find percentage (between the mean) for both options using Table A

3) Convert percentage to Probability for both options

4) Add Probability for both options together

**CHAPTER 5**

Mean, median and mode coincide

43.32% between mean and z

6.68% beyond z