Name: Date:
Ms. D’Amato Block:
Chapter 6: The Standard Deviation as a Ruler and the Normal Model
The Standard Deviation as a Ruler
· The trick in comparing very different-looking values is to use standard deviations as our rulers.
· The standard deviation tells us how the whole collection of values varies, so it’s a natural ruler for comparing an individual to a group.
· As the most common measure of variation, the standard deviation plays a crucial role in how we look at data.
Shifting Data
· Adding (or subtracting) a constant amount to each value just adds (or subtracts) the same constant to (from) the mean. This is true for the median and other measures of position too.
· In general, adding a constant to every data value adds the same constant to measures of center and percentiles, but leaves measures of spread unchanged.
· The following histograms show a shift from men’s actual weights to kilograms above recommended weight:
Example 1: In 1995 the Educational Testing Service (ETS) adjusted the scores of SAT tests. Before ETS recentered the SAT Verbal test, the mean of all test scores was 450.
a.) How would adding 50 points to each score affect the mean?
b.) The standard deviation was 100 points. What would the standard deviation be after adding 50 points?
c.) Suppose we drew boxplots of test takers’ scores a year before and a year after the re-centering. How would the boxplots of the two years differ?
Rescaling Data
· When we divide or multiply all the data values by any constant value, both measures of location (e.g., mean and median) and measures of spread (e.g., range, IQR, standard deviation) are divided and multiplied by the same value.
· The men’s weight data set measured weights in kilograms. If we want to think about these weights in pounds, we would rescale the data:
Example 2: A company manufactures wheels for roller blades. The diameter of the wheels has a mean of 3 inches and a standard deviation of 0.1 inches. Because so many of its customers use the metric system, the company decided to report their production statistics in millimeters (1 inch = 25.4 mm). They report that the standard deviation is now 2.54 mm. A corporate executive is worried about this increase in variation. Should they be concerned? Explain.
The 68-95-99.7 Rule
· There is a model that shows up over and over in Statistics and this model is called the
(You may have heard of “bell-shaped curves” or Normal
Distribution).
· Normal models are appropriate for distributions whose shapes are and
.
· There is a Normal model for every possible combination of mean and standard deviation.
o We write to represent a Normal model with a mean of μ and a standard deviation of σ.
· We use Greek letters because this mean and standard deviation do not come from data—they
are numbers (called ) that specify the model.
· Summaries of data, like the sample mean and standard deviation, are written with Latin
letters. Such summaries of data are called .
· Normal models give us an idea of how extreme a value is by telling us how likely it is to find one that far from the mean.
· We can find these numbers precisely, but until then we will use a simple rule that tells us a lot about the Normal model…
· It turns out that in a Normal model:
o about of the values fall within one standard deviation of the mean;
o about of the values fall within two standard deviations of the mean; and,
o about (almost all!) of the values fall within three standard deviations of the mean.
· The following shows what the 68-95-99.7 Rule tells us:
Example 3: Suppose it takes you 20 minutes, on average, to drive to school, with a standard deviation of 2 minutes. Suppose a Normal model is appropriate for the distributions of driving times.
A] How often will you arrive at school in less than 22 minutes?
B] How often will it take you more than 24 minutes?
C] How often will it take you more than 16 minutes but less than 20 minutes?
Example 4: A machine fills 12 ounce Potato Chip bags. It places chips in the bags. Not all bags weigh exactly 12 ounces. The weight of the chips placed is normally distributed with a mean of 12.4 ounces and with a standard deviation of 0.2 ounces.
The company has asked you to determine the following probabilities to aid in consumer relations concerning the weight of the bags purchased.
A] If you purchase a bag filled by this dispenser, what is the likelihood it has less than 12 ounces?
B] If you purchase a bag filled by this dispenser, what is the likelihood it has more than 12 ounces?
Finding Normal Percentiles by Hand
· When a data value doesn’t fall exactly 1, 2, or 3 standard deviations from the mean, we can look
it up in a table of Normal percentiles.
Finding Normal Percentiles Using Technology
· Many calculators and statistics programs have the ability to find normal percentiles for us.
Example 5: A machine fills 12 ounce Potato Chip bags. It places chips in the bags. Not all bags weigh exactly 12 ounces. The weight of the chips placed is normally distributed with a mean of 12.4 ounces and with a standard deviation of 0.2 ounces.
The company has asked you to determine the following probabilities to aid in consumer relations concerning the weight of the bags purchased.
A] If you purchase a bag filled by this dispenser, what is the likelihood it has less than 12.1 ounces?
B] If you purchase a bag filled by this dispenser, what is the likelihood it has more than 12.5 ounces?
C] What weight of the bag is represented by the 84th percentile?
Example 6: Based on the Normal model N(100, 16) describing IQ scores, answer the following questions.
Drawing:
A] What percent of people’s IQs would you expect to be over 80?
B] What percent of people’s IQs would you expect to be under 90?
C] What percent of people’s IQs would you expect to be between 112 and 132?
D] What cutoff value bounds the highest 5% of all IQs?
E] What cutoff value bounds the lowest 30% of all IQs?
F] What IQ would represent the 50th percentile?
Standardizing with z-scores
· We compare individual data values to their mean, relative to their standard deviation using the following formula:
· We call the resulting values , denoted as z. They can
also be called .
· Standardized values have .
· z-scores measure the distance of each data value from the mean in standard deviations.
· A negative z-score tells us that the data value is the mean, while a positive
z-score tells us that the data value is the mean.
· Standardized values have been converted from their original units to the standard statistical unit of standard deviations from the mean.
· Thus, we can compare values that are measured on different scales, with different units, or from different populations.
· Standardizing data into z-scores the data by subtracting the mean and
the values by dividing by their standard deviation.
o Standardizing into z-scores does not change the of the distribution.
o Standardizing into z-scores changes the by making the mean .
o Standardizing into z-scores changes the by making the standard
deviation .
When is a z-score BIG?
· A z-score gives us an indication of how unusual a value is because it tells us how far it is from
.
· Remember that a negative z-score tells us that the data value is the mean,
while a positive z-score tells us that the data value is the mean.
· The larger a z-score is (negative or positive), the more it is.
· There is no universal standard for z-scores, but the Normal model provides a measure of how extreme a z-score is.
· When we standardize Normal data, we still call the standardized value a z-score, and we write
· Once we have standardized, we need only one model:
o The N(0, 1) model is called the (or the
standard Normal distribution).
Example 7: Calculate the z-score of each data point if we have a data set where μ = 8 and σ = 2.
A] y = 12 B] y = 6
Example 8: What percent of a standard Normal model is found in each region? Draw a picture.
a. z < -0.69
Answer:
b. z > -2.75
Answer:
c. -0.7 < z < 1.2
Answer:
Example 9: In a standard Normal model, what are the “z” values that describe the following percents?
A] the lowest 28%
B] the highest 43%
Example 10: Your statistics teacher has announced that the lower of your two tests will be dropped. You got a 90 on test 1 and an 80 on test 2. You’re all set to drop the 80 until she announces that she grades “on a curve.” She standardized the scores in order to decide which test is the lower one. If the mean on the first test is 88 with a standard deviation of 4 and the mean on the second was a 75 with a standard deviation of 5. Which one will be dropped?
Example 11:
The average apple has a diameter of 3.25 inches with a standard deviation of .5 inches. The average orange has a diameter of 4.5 inches and has a standard deviation of 1 inch. If I have an apple with a diameter of 4 inches and an orange with a diameter of 5.5 inches, which fruit is largest compared to others of its kind? How much larger (What percentage?)?
APPLES ORANGES
μ : μ :
σ : σ :
y : y :
Draw the picture: Draw the picture:
Work with Formula: Work with Formula:
Use Calculator to determine percentage: Use Calculator to determine percentage: