math 110 sec 14-4 lecture: the normal distribution the normal distribution describes many real-life...

MATH 110 Sec 14-4 Lecture: The Normal Distribution

The normal distribution describes many real-life data sets.

The normal distribution describes many real-life data sets.The histogram shown gives an idea of

the shape of a normal distribution.

The normal distribution describes many real-life data sets.Although the normal distribution is a continuous distribution whose graph is a smooth curve, an appropriate histogram can

give a very good approximation to the actual normal graph.

Properties of the Normal Distribution

Properties of the Normal Distribution is the mean and is the standard deviation of the distribution.

1. A normal curve is bell-shaped. is the mean and is the standard deviation of the distribution.

1. A normal curve is bell-shaped.2. The highest point on the curve is at the mean.

is the mean and is the standard deviation of the distribution.

1. A normal curve is bell-shaped.2. The highest point on the curve is at the mean.3. The mean, median and mode are equal.

1. A normal curve is bell-shaped.2. The highest point on the curve is at the mean.3. The mean, median and mode are equal.4. The curve is symmetric with respect to its mean.

1. A normal curve is bell-shaped.2. The highest point on the curve is at the mean.3. The mean, median and mode are equal.4. The curve is symmetric with respect to its mean.5. The total area under the curve is 1.

1. A normal curve is bell-shaped.2. The highest point on the curve is at the mean.3. The mean, median and mode are equal.4. The curve is symmetric with respect to its mean.5. The total area under the curve is 1.6. Roughly 68% of the data values are within 1 standard deviation of the mean, 95% are within 2 standard deviations of the mean and 99.7% are within 3 standard deviations of the mean.

The 68-95-99.7 Rule for Normal Distributions

MATH 110 Sec 14-4 Lecture: The Normal Distribution The distribution of scores of 1000 students on an intelligence test is a

normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall between 425 & 475?

The 68-95-99.7 Rule68% of the values are within 1 std. deviation of the mean,

95% are within 2 standard deviations of the mean and 99.7% are with 3 standard deviations of the mean.

The distribution of scores of 1000 students on an intelligence test is a normal distribution with mean 450 and standard deviation of 25.

What percent of scores would be expected to fall between 425 & 475?

425 475

The shaded area gives the probability of a score falling in the 425 – 475 range.

The 68-95-99.7 Rule 68% of the values are within 1 std. deviation of the mean,

Here (mean) and (std deviation)

450425 475

450 – 425 = 2525

450425 475

475 – 450 = 2525 25

450425 475

25 25But 25 is the standard deviation.

425 475

25 25But 25 is the standard deviation.The interval shown consists of all scores

within 1 standard deviation of the mean.

450The shaded area gives the probability of

a score falling in the 425 – 475 range.

425 475

The 68-95-99.7 Rule tells us that 68% of the scores are within 1 standard deviation of the mean.

The interval shown consists of all scores within 1 standard deviation of the mean.

425 475

68%The interval shown consists of all scores within 1 standard deviation of the mean.

425 475

About 68% of scores

should fall between

425 & 475.

425 475

About 68% of scores

should fall between

425 & 475.

How many of the 1000 scores would we expect to be between 425 & 475?

425 475

If there are 1000 scores, we would expect about 68% of them to be between 425 and 475.

About 68% of scores

should fall between

425 & 475.

425 475

If there are 1000 scores, we would expect about 68% of them to be between 425 and 475.1000 (68% )=1000 (0.68 )=680

About 68% of scores

should fall between

425 & 475.

425 475

450So, we expect about 680 scores to be in

the 425 – 475 range.

If there are 1000 scores, we would expect about 68% of them to be between 425 and 475.1000 (68% )=1000 (0.68 )=680

About 68% of scores

should fall between

425 & 475.

normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall above 500?

This is exactly the same distribution as before.

What percent of scores would be expected to fall above 500?

This is exactly the same distribution as before.Only the question has changed.

The shaded area is the probability of a score being above 500.

First we need to find how many standard deviations 500 is from the mean.

Only the question has changed.

500 – 450 = 5050

But 25 is the standard deviation.

500 – 450 = 5050

But 25 is the standard deviation.So 500 is 2 standard deviations () from the mean.

So 500 is 2 standard deviations () from the mean.

The 68-95-99.7 Rule tells us that 95% of the scores are within 2 standard deviations of the mean.

That leaves 5% to be split between the

two tails.

Half of 5% is 2.5%.So the orange shaded area is 2.5%.

two tails.

2.5%2.5%

two tails.

2.5%2.5%

About 2.5% of scores should be above 500.

two tails.

2.5%2.5%

How many of the 1000 scores would we expect to be above 500?

two tails.

2.5%2.5%

We expect about 2.5% of them to be above 500.

two tails.

2.5%2.5%

1000 (2.5% )=1000 (0.025 )=25

two tails.

2.5%2.5%

So, we expect about 25 scores to be above 500.1000 (2.5% )=1000 (0.025 )=25

MATH 110 Sec 14-4 Lecture: The Normal Distribution This procedure can be somewhat simplified by noticing that the number

of standard deviations of the interval endpoints from the mean determines the percentages under the normal curve.

𝜇+𝜎𝜇−𝜎 𝜇

of scores

When endpoints are 1 standard deviation from the mean

of scores34% 34%

of scores34% 34%16% 16%

When endpoints are 2 standard deviations from the mean

𝜇+2𝜎𝜇−2𝜎 𝜇

of scores

of scores47.5% 47.5%

of scores47.5%2.5% 2.5%47.5%

of scores

of scores49.85% 49.85%

of scores49.85%0 .15% 49.85% 0 .15%

of scores47.5%2.5% 2.5%47.5%

of scores34% 34%16% 16%

SUMMARY

How each percentage

of the 68-95-99.7 rule breaks down

underneaththe Normal Curve

normal distribution with mean 450 and standard deviation of 25. What percent of scores would be expected to fall below 375?

What percent of scores would be expected to fall below 375?

The shaded area gives the probability of a score falling below 375.

450 – 375 = 7575

450 – 375 = 75

But 25 is the standard deviation.

450 – 375 = 75

But 25 is the standard deviation.So 375 is 3 standard deviations (75) from the mean.

So 375 is 3 standard deviations (75) from the mean.

The 68-95-99.7 Rule tells us that 99.7% of the scores are within 3 standard deviations of the mean.

This time, let’s take

advantage of the summary

sheet we developed.

of scores49.85%0 .15% 49.85% 0 .15%

of scores47.5%2.5% 2.5%47.5%

of scores34% 34%16% 16%

SUMMARY

How each percentage

of scores49.85%0 .15% 49.85% 0 .15%

of scores47.5%2.5% 2.5%47.5%

of scores34% 34%16% 16%

SUMMARY

How each percentage

We figured out that 375 is 3 standard deviations from the mean, so the

bottom graph is the one that we need.

sheet we developed.

of scores49.85%0 .15% 49.85% 0 .15%

sheet we developed.

of scores49.85%0 .15% 49.85% 0 .15%

sheet we developed.

of scores49.85%0 .15% 49.85% 0 .15%

sheet we developed.

of scores49.85%0 .15% 49.85% 0 .15%

So the orange shaded area is 0.15%.

sheet we developed.

of scores49.85%0 .15% 49.85% 0 .15%

So the orange shaded area is 0.15%.

So, we expect about 0.15%of the scores to be below 375.

normal distribution with mean 150 and standard deviation of 10. How many scores do we expect to fall between 150 and 180?

Given what we have

learned, let’s try to solve

this as efficiently as

possible.

Given what we have

possible.

Given what we have

possible.First, we must decide which part of the

‘68-95-99.7 Rule’ applies.

Given what we have

possible.

To decide the percent to use, we must find the number of standard

deviations there are between180 and the mean (150).

First, we must decide which part of the

Given what we have

possible.180−150=30180−150=30180−150=30

180−150=30 First, we must decide which part of the

Given what we have

possible.

180−150=30

30 (10+10+10) is3 standard deviations

Given what we have

possible.

180−150=30So, 180 is 3 standard deviations

from the mean.

30 (10+10+10) is3 standard deviations

Given what we have

possible.

from the mean.

Given what we have

possible.

from the mean.

So, the3 standard

deviation part of the

of scores49.85%0 .15% 49.85% 0 .15%

Given what we have

possible.

from the mean.

So, the3 standard

of scores49.85%0 .15% 49.85% 0 .15%

150 180

Given what we have

possible.

from the mean.

So, the3 standard

of scores49.85%0 .15% 49.85% 0 .15%

150 180

49.85% is the percent between 150 and 180.

Given what we have

possible.

from the mean.

So, the3 standard

of scores49.85%0 .15% 49.85% 0 .15%

150 180

49.85% is the percent between 150 and 180.2000 (49.85% )=2000 (0.4985 )=997

Given what we have

possible.

from the mean.

So, the3 standard

of scores49.85%0 .15% 49.85% 0 .15%

150 180

49.85% is the percent between 150 and 180.2000 (49.85% )=2000 (0.4985 )=997 So, we expect about 997 scores to

be between 150 and 180.

MATH 110 Sec 14-4 Lecture: The Normal Distribution A normal distribution has a mean of 36 and a standard deviation of 5.

What percentage of values do we expect to be below 26?

What percentage of values do we expect to be below 26?Let’s use this exercise to try to get the answer while showing even less work.

Step 1. Find the difference between 26 and 36 (mean).

Step 1. Find the difference between 26 and 36 (mean).𝟑𝟔−𝟐𝟔=𝟏𝟎

Step 2. Express that difference in terms of standard deviations.

𝟑𝟔−𝟐𝟔=𝟏𝟎

Step 2. Express that difference in terms of standard deviations.Standard deviation is 5 so 10 is 2 standard deviations from the mean.

𝟑𝟔−𝟐𝟔=𝟏𝟎

Step 3. Use the percent diagram for the ‘2 standard deviation’ case.

𝟑𝟔−𝟐𝟔=𝟏𝟎

Step 3. Use the percent diagram for the ‘2 standard deviation’ case.When endpoints are 2 standard

deviations from the mean

of scores47.5%2.5% 2.5%47.5%

𝟑𝟔−𝟐𝟔=𝟏𝟎

of scores47.5%2.5% 2.5%47.5%

Step 4. Find the percent that goes with ‘below 26’.

𝟑𝟔−𝟐𝟔=𝟏𝟎

of scores47.5%2.5% 2.5%47.5%

Step 4. Find the percent that goes with ‘below 26’. 26

𝟑𝟔−𝟐𝟔=𝟏𝟎

of scores47.5%2.5% 2.5%47.5%

𝟑𝟔−𝟐𝟔=𝟏𝟎

of scores47.5%2.5% 2.5%47.5%

About 2.5% of values will be below 26.Step 5. Answer:

math 110 sec 14-4 lecture: the normal distribution the normal distribution describes many real-life...

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