get out your density curve ws! you will be able to describe a normal curve. you will be able to find...
TRANSCRIPT
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.
Todayβs Objectives:
Normal Curves
The Normal curves are one of the most common types of density curve.
All Normal curves are... β’ Symmetricβ’ Single-peakedβ’ Bell-shaped
Normal Curves
Normal curves describe Normal distributions.
Normal distributions are anything but βnormal.β They play a huge role in statistics.
Capitalize the βnβ in normal when referring to a Normal distribution or curve.
Why Important?
Normal distributions are important in statistics becauseβ¦
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 populationsβ¦
Why Important?
Normal distributions are important in statistics becauseβ¦
2. Normal distributions are good approximations to the results of many kinds of chance outcomes.
Ex. tossing a coin many times, rolling a dieβ¦
Why Important?
Normal distributions are important in statistics becauseβ¦
2. Most importantly, Normal distributions are the basis for many statistical inference procedures.
Inferenceβthe process of arriving at some conclusion that possesses some degree of probability relative to the premises
Normal Curvesβ’ Symmetric, single-peaked, and bell-shaped.
β’ Tails fall off quickly, so do not expect outliers.
β’ 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 68β95β99.7 Rule because these values describe the distribution.
The empirical rule applies only to NORMAL DISTRIBUTIONS!!!!!
The Empirical Rule
The Empirical Rule states that
β’ Approximately 68% of the data values fall within one standard deviation of the mean
β’ Approximately 95% of the data values fall within 2 standard deviations of the mean
β’ 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
β1π 1ππ₯
The Empirical Rule
of the observations fall within standard deviation around the mean.
95% of data
β1π 1ππ₯ 2πβ2π
The Empirical Rule
of the observations fall within standard deviation around the mean.
99.8% of data
β1π 1ππ₯ 2πβ2π 3πβ3π
The Empirical Rule99.8% of data
95% of data
68% of data
π₯β3ππ₯β2ππ₯β1ππ₯ π₯+1ππ₯+2ππ₯+3π
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%
π₯β3ππ₯β2ππ₯β1ππ₯π₯+1ππ₯+2ππ₯+3π
Examples1. What percent of the data scores
were above a 76?
13.5% 34% 34% 13.5%2.4% 2.4% .1%.1%
π₯β3ππ₯β2ππ₯β1ππ₯π₯+1ππ₯+2ππ₯+3π
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%
π₯β3ππ₯β2ππ₯β1ππ₯π₯+1ππ₯+2ππ₯+3π
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%
π₯β3ππ₯β2ππ₯β1ππ₯π₯+1ππ₯+2ππ₯+3π
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%
π₯β3ππ₯β2ππ₯β1ππ₯π₯+1ππ₯+2ππ₯+3π
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%
π₯β3ππ₯β2ππ₯β1ππ₯π₯+1ππ₯+2ππ₯+3π
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%
π₯β3ππ₯β2ππ₯β1ππ₯π₯+1ππ₯+2ππ₯+3π
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%
π₯β3ππ₯β2ππ₯β1ππ₯π₯+1ππ₯+2ππ₯+3π
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%
π₯β3ππ₯β2ππ₯β1ππ₯π₯+1ππ₯+2ππ₯+3π
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%
π₯β3ππ₯β2ππ₯β1ππ₯π₯+1ππ₯+2ππ₯+3π
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%
π₯β3ππ₯β2ππ₯β1ππ₯π₯+1ππ₯+2ππ₯+3π
53 62.5 72 81.524.5 34 43.5