Get out your Density Curve WS!

You will be able to describe a Normal curve.

You will be able to find percentages and standard deviations based on a Normal curve.

Today’s Objectives:

Normal Curves

The Normal curves are one of the most common types of density curve.

All Normal curves are... • Symmetric• Single-peaked• Bell-shaped

Normal Curves

Normal curves describe Normal distributions.

Normal distributions are anything but “normal.” They play a huge role in statistics.

Capitalize the “n” in normal when referring to a Normal distribution or curve.

Why Important?

Normal distributions are important in statistics because…

1. Normal distributions are good descriptions for some distributions of real data.

Ex. Scores on tests, repeated careful measures of the same quantity, characteristics of biological populations…

Why Important?

Normal distributions are important in statistics because…

2. Normal distributions are good approximations to the results of many kinds of chance outcomes.

Ex. tossing a coin many times, rolling a die…

Why Important?

Normal distributions are important in statistics because…

2. Most importantly, Normal distributions are the basis for many statistical inference procedures.

Inference—the process of arriving at some conclusion that possesses some degree of probability relative to the premises

Normal Curves• Symmetric, single-peaked, and bell-shaped.

• Tails fall off quickly, so do not expect outliers.

• Mean, Median, and Mode are all located at the peak in the center of the curve.

Mean Median Mode

Normal Curves

The mean fixes the center of the curve and the standard deviation determines its shape.

The standard deviation fixes the spread of a Normal curve. Remember, spread tells us how much a data sample is spread out or scattered.

Normal CurvesThe mean and the standard deviation completely specifies the curve.Changing the mean changes its location on the axis.Changing the standard deviation changes the shape of a Normal curve.

The Empirical RuleAlso known as the 68—95—99.7 Rule because these values describe the distribution.

The empirical rule applies only to NORMAL DISTRIBUTIONS!!!!!

The Empirical Rule

The Empirical Rule states that

• Approximately 68% of the data values fall within one standard deviation of the mean

• Approximately 95% of the data values fall within 2 standard deviations of the mean

• Approximately 99.8% of the data values fall within 3 standard deviations of the mean

The Empirical Rule

of the observations fall within standard deviation around the mean.

68% of data

−1𝑆 1𝑆𝑥

The Empirical Rule

of the observations fall within standard deviation around the mean.

95% of data

−1𝑆 1𝑆𝑥 2𝑆−2𝑆

The Empirical Rule

of the observations fall within standard deviation around the mean.

99.8% of data

−1𝑆 1𝑆𝑥 2𝑆−2𝑆 3𝑆−3𝑆

The Empirical Rule99.8% of data

95% of data

68% of data

𝑥−3𝑆𝑥−2𝑆𝑥−1𝑆𝑥 𝑥+1𝑆𝑥+2𝑆𝑥+3𝑆

ExamplesAssume you have a normal distribution of test scores with a mean of 82 and a standard deviation of 6.

13.5% 34% 34% 13.5%2.4% 2.4% .1%.1%

𝑥−3𝑆𝑥−2𝑆𝑥−1𝑆𝑥𝑥+1𝑆𝑥+2𝑆𝑥+3𝑆

Examples1. What percent of the data scores

were above a 76?

13.5% 34% 34% 13.5%2.4% 2.4% .1%.1%

𝑥−3𝑆𝑥−2𝑆𝑥−1𝑆𝑥𝑥+1𝑆𝑥+2𝑆𝑥+3𝑆

82 88 94 10064 70 76

Examples2. 68% of the data fall between what

two scores?

13.5% 34% 34% 13.5%2.4% 2.4% .1%.1%

𝑥−3𝑆𝑥−2𝑆𝑥−1𝑆𝑥𝑥+1𝑆𝑥+2𝑆𝑥+3𝑆

82 88 94 10064 70 76

Examples3. What percent of the data scores fall between 70 and 100?

13.5% 34% 34% 13.5%2.4% 2.4% .1%.1%

𝑥−3𝑆𝑥−2𝑆𝑥−1𝑆𝑥𝑥+1𝑆𝑥+2𝑆𝑥+3𝑆

82 88 94 10064 70 76

Examples4. How many standard deviations away from the mean is 88, and in which direction?

13.5% 34% 34% 13.5%2.4% 2.4% .1%.1%

𝑥−3𝑆𝑥−2𝑆𝑥−1𝑆𝑥𝑥+1𝑆𝑥+2𝑆𝑥+3𝑆

82 88 94 10064 70 76

Examples5. How many standard deviations away from the mean is 64, and in which direction?

13.5% 34% 34% 13.5%2.4% 2.4% .1%.1%

𝑥−3𝑆𝑥−2𝑆𝑥−1𝑆𝑥𝑥+1𝑆𝑥+2𝑆𝑥+3𝑆

82 88 94 10064 70 76

ExamplesA charity puts on a relay race to raise money. The times of the finishes are normally distributed with a mean of 53 minutes and a standard deviation of 9.5 minutes.

13.5% 34% 34% 13.5%2.4% 2.4% .1%.1%

𝑥−3𝑆𝑥−2𝑆𝑥−1𝑆𝑥𝑥+1𝑆𝑥+2𝑆𝑥+3𝑆

Examples1. What percent of the data times were

between 34 and 81.5 minutes?

13.5% 34% 34% 13.5%2.4% 2.4% .1%.1%

𝑥−3𝑆𝑥−2𝑆𝑥−1𝑆𝑥𝑥+1𝑆𝑥+2𝑆𝑥+3𝑆

53 62.5 72 81.524.5 34 43.5

Examples2. 95% of the data fall between what

two times?

13.5% 34% 34% 13.5%2.4% 2.4% .1%.1%

𝑥−3𝑆𝑥−2𝑆𝑥−1𝑆𝑥𝑥+1𝑆𝑥+2𝑆𝑥+3𝑆

53 62.5 72 81.524.5 34 43.5

Examples3. What percent of the data times

were below 72 minutes?

13.5% 34% 34% 13.5%2.4% 2.4% .1%.1%

𝑥−3𝑆𝑥−2𝑆𝑥−1𝑆𝑥𝑥+1𝑆𝑥+2𝑆𝑥+3𝑆

53 62.5 72 81.524.5 34 43.5

Examples4. How many standard deviations away from the mean is 34 and in which direction?

13.5% 34% 34% 13.5%2.4% 2.4% .1%.1%

𝑥−3𝑆𝑥−2𝑆𝑥−1𝑆𝑥𝑥+1𝑆𝑥+2𝑆𝑥+3𝑆

53 62.5 72 81.524.5 34 43.5

Homework

Normal Curve WorksheetDue Monday