AP Statistics The Standard Deviation as a Ruler and the Normal Model Chapter 6

Learning Goals 1.Understand how adding (subtracting) a constant or multiplying (dividing) by a constant changes the center and/or spread of a variable. 2.Understand that standardizing uses the standard deviation as a ruler. 3.Know how to calculate the z-score of an observation and what it means. 4.Know how to compare values of two different variables using their z-scores. 5.Recognize when a Normal model is appropriate.

Learning Goals 6.Recognize when standardization can be used to compare values. 7.Be able to use Normal models and the Rule to estimate the percentage of observations falling within 1, 2, or 3 standard deviations of the mean. 8.Know how to find the percentage of observations falling below any value in a Normal model using a Normal table or appropriate technology. 9.Know how to check whether a variable satisfies the Nearly Normal Condition by making a Normal Probability plot or histogram.

Learning Goal 1 Understand how adding (subtracting) a constant or multiplying (dividing) by a constant changes the center and/or spread of a variable.

Learning Goal 1: Linear Transformation of Data Linear transformation – Shifting (moving left or right) the data or Rescaling (making the size larger or smaller) the data. – Changes the original variable x into the new variable x new given by x new = a + bx Adding the constant a shifts all values of x upward (right) or downward (left) by the same amount. Multiplying by the positive constant b changes the size of the values or rescales the data.

Learning Goal 1: Shifting 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.

Learning Goal 1: Shifting Data Example: Adding a Constant Given the data: 2, 4, 6, 8, 10 – Center: mean = 6, median = 6 – Spread: s = 3.2, IQR = 6 Add a constant 5 to each value, new data 7, 9, 11, 13, 15 – New center: mean = 11, median = 11 – New spread: s = 3.2, IQR = 6 Effects of adding a constant to each data value – Center increases by the constant 5 – Spread does not change – Shape of the distribution does not change

Learning Goal 1: Example - Subtracting a Constant The following histograms show a shift from men’s actual weights to kilograms above recommended weight. 1.No change in shape 2.No change in spread 3.Shift by 74 Shift Down

Learning Goal 1: Rescaling Data Rescaling data: – When we divide or multiply all the data values by any constant value, all measures of position (such as the mean, median and percentiles) and measures of spread (such as the range, IQR, and standard deviation) are divided and multiplied by that same constant value.

Learning Goal 1: Rescaling Example: Multiplying by a Constant Given the data: 2, 4, 6, 8, 10 – Center: mean = 6, median = 6 – Spread: s = 3.2, IQR = 6 Multiple a constant 3 to each value, new data: 6, 12, 18, 24, 30 – New center: mean = 18, median = 18 – New spread: s = 9.6, IQR = 18 Effects of multiplying each value by a constant – Center increases by a factor of the constant (times 3) – Spread increases by a factor of the constant (times 3) – Shape of the distribution does not change

Learning Goal 1: Example - Rescaling Data 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: 1.No change in shape 2.Increase in spread 3.Shift by to the right Size (spread) increases Shifts up

Learning Goal 1: Summary of Linear Transformations Multiplying each observation by a positive number b multiples both measures of center (mean and median) and measures of spread (IQR and standard deviation) by b. Adding the same number a (either positive or negative) to each observation adds a to measures of center and to quartiles, but does not change measures of spread. Linear transformations do not change the shape of a distribution.

Learning Goal 1: Summary of Linear Transformations Linear transformations do not affect the shape of the distribution of the data. -for example, if the original data is right- skewed, the transformed data is right- skewed. Example, changing the units of data from minutes to seconds (multiplying by 60 sec/min). Shape remains the same

Learning Goal 1: Example (a) Suppose that each member of the team receives a $100,000 bonus for winning the NBA championship. How will this affect the center and spread? $100,000 = $0.1 mil Los Angeles Laker’s Salaries (2000)

Learning Goal 1: Example - Solution

Learning Goal 1: Example (b) Each player is offered a 10% increase base salary. How will this affect the center and spread?

Learning Goal 1: Example - Solution

Learning Goal 1: Your Turn Maria measures the lengths of 5 cockroaches that she finds at school. Here are her results (in inches): a.Find the mean and standard deviation of Maria’s measurements (use calc). b.Maria’s science teacher is furious to discover that she has measured the cockroach lengths in inches rather than centimeters (There are 2.54 cm in 1 inch). She gives Maria two minutes to report the mean and standard deviation of the 5 cockroaches in centimeters. Find the mean and standard deviation in centimeters.

Learning Goal 1: Class Problem We have a company with employees with the following salaries: What is the mean and standard deviation of the company salaries? mean = _________st dev = _________ 2.Suppose we give everyone a $500 raise. What is the new mean and standard deviation? mean = _________st dev = _________ 3.Suppose we have to cut everyone’s pay by $500 due to the economy. What is the mean and standard deviation now? mean = _________st dev = _________

Learning Goal 1: Class Problem (continued) 4.Suppose we give everyone a 30% raise. What is the new mean and standard deviation? mean = _________st dev = _________ 5.Suppose we cut everyone’s pay by 7%. What is the new mean and standard deviation? mean = _________st dev = _________

Learning Goal 2 Understand that standardizing uses the standard deviation as a ruler.

Learning Goal 2: Comparing Apples to Oranges In order to compare data values with different units (apples and oranges), we need to make sure we are using the same scale. The trick is to look at how the values deviate from the mean. Look at whether the data point is above or below the mean, and by how much. Standard deviation measures that, the deviation of the data values from the mean.

Learning Goal 2: The Standard Deviation as a Ruler As the most common measure of variation, the standard deviation plays a crucial role in how we look at data. The trick in comparing very different-looking values is to use standard deviations as our ruler. 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.

Learning Goal 2: The Standard Deviation as a Ruler We compare individual data values to their mean, relative to their standard deviation using the following formula: We call the resulting values standardized values, denoted as z. They can also be called z-scores.

Learning Goal 3 Know how to calculate the z-score of an observation and what it means.

Learning Goal 3: Standardizing with z-scores When x is larger than the mean, z is positive. When x is smaller than the mean, z is negative.

Learning Goal 3: Standardizing with z-scores A z-score puts values on a common scale. Standardized values have no units. z-scores measure the distance of each data value is from the mean in standard deviations. z-scores farther from 0 are more extreme. z-scores beyond -2 or 2 are considered unusual. A negative z-score tells us that the data value is below the mean, while a positive z-score tells us that the data value is above the mean.

Learning Goal 3: Standardizing with z-scores Gives a common scale. – We can compare two different distributions with different means and standard deviations. Z-Score tells us how many standard deviations the observation falls away from the mean. Observations greater than the mean are positive when standardized and observations less than the mean are negative. This Z-Score tells us it is 2.15 Standard Deviations from the mean 2.15 SD Z=-2.15

Learning Goal 3: Standardizing with z-scores - Example

Learning Goal 3: Standardizing with z-scores – Your Turn Bob is 64 inches tall. The heights of men are unimodal symmetric with a mean of 69 inches and standard deviation of 2.5 inches. How does Bob’s height compare to other men.

Learning Goal 3: Problem Which one of the following is a FALSE statement about a standardized value (z-score)? a)It represents how many standard deviations an observation lies from the mean. b)It represents in which direction an observation lies from the mean. c)It is measured in the same units as the variable.

Learning Goal 3: Problem

Learning Goal 4 Know how to compare values of two different variables using their z-scores.

Learning Goal 4: Benefits of Standardizing Standardized values have been converted from their original units to the standard statistical unit of standard deviations from the mean (z-score). Thus, we can compare values that are measured on different scales, with different units, or from different populations.

Learning Goal 4: Standardizing – Comparing Distributions The men’s combined skiing event in the in the winter Olympics consists of two races: a downhill and a slalom. In the 2006 Winter Olympics, the mean slalom time was seconds with a standard deviation of seconds. The mean downhill time was seconds with a standard deviation of seconds. Ted Ligety of the U.S., who won the gold medal with a combined time of seconds, skied the slalom in seconds and the downhill in seconds. On which race did he do better compared with the competition?

Learning Goal 4: Standardizing – Comparing Distributions

Learning Goal 4: Standardizing – Your Turn Timmy gets a 680 on the math of the SAT. The SAT score distribution is Unimodal symmetric with a mean of 500 and a standard deviation of 100. Little Jimmy scores a 27 on the math of the ACT. The ACT score distribution is unimodal symmetric with a mean of 18 and a standard deviation of 6. Who does better? (Hint: standardize both scores then compare z-scores)

Learning Goal 4: Standardizing – Your Turn

Learning Goal 4: Standardizing – Class Problem A town’s January high temp averages 36 ̊F with a standard deviation of 10, while in July, the mean high temp is 74 ̊F with a standard deviation of 8. In which month is it more unusual to have a day with a high temp of 55 ̊F?

Learning Goal 4: Standardizing – Combining z-scores Because z-scores are standardized values, measure the distance of each data value from the mean in standard deviations and have no units, we can also combine z-scores of different variables.

Learning Goal 4: Standardizing – Example: Combining z-scores In the 2006 Winter Olympics men’s combined event, Ted Ligety of the U.S. won the gold medal with a combined time of seconds. Ivica Kostelic of Croatia skied the slalom in seconds and the downhill in seconds, for a combined time of seconds. Considered in terms of combined z- scores, who should have won the gold medal?

Learning Goal 4: Solution Ted Ligety: Combined z-score: Ivica Kostelic: Combined z-score: Using standardized scores, overall Kostelic did better and should have won the gold.

Learning Goal 4: Combining z-scores - Your Turn The distribution of SAT scores has a mean of 500 and a standard deviation of 100. The distribution of ACT scores has a mean of 18 and a standard deviation of 6. Jill scored a 680 on the math part of the SAT and a 30 on the ACT math test. Jack scored a 740 on the math SAT and a 27 on the math ACT. Who had the better combined SAT/ACT math score?

Learning Goal 5 Recognize when a Normal model is appropriate.

Learning Goal 5: Smooth Curve Sometimes the overall pattern of a histogram is so regular that it can be described by a Smooth Curve. This can help describe the location of individual observations within the distribution.

Learning Goal 5: Smooth Curve The distribution of a histogram depends on the choice of classes, while with a smooth curve it does not. Smooth curve is a mathematical model of the distribution. – How? The smooth curve describes what proportion of the observations fall in each range of values, not the frequency of observations like a histogram. Area under the curve represents the proportion of observations in an interval. The total area under the curve is 1.

Learning Goal 5: Mathematical Model Histogram of a sample with the smoothed, density curve describing theoretically the population.

Learning Goal 5: The Normal Model There is no universal standard for z- scores, but there is a model that shows up over and over in Statistics. This model is called the Normal Model (You may have heard of “bell- shaped curves.”). Normal models are appropriate for distributions whose shapes are unimodal and roughly symmetric. These distributions provide a measure of how extreme a z-score is.

Learning Goal 5: The Normal Model Normal Model: One Particular class of distributions or model. 1.Symmetric 2.Single Peaked 3.Bell Shaped All have the same overall shape.

Learning Goal 5: The Normal Model or Normal Distribution The normal distribution is considered the most important distribution in all of statistics. It is used to describe the distribution of many natural phenomena, such as the height of a person, IQ scores, weight, blood pressure etc. 9-50

Learning Goal 5: The Normal Distribution The mathematical equation for the normal distribution is given below: where e  2.718,   3.14,  = population mean, and  = population standard deviation. Not required to know.

Learning Goal 5: Properties of the Normal Distribution When this equation is graphed for a given  and , a continuous, bell- shaped, symmetric graph will result. Thus, we can display an infinite number of graphs for this equation, depending on the value of  and . In such a case, we say we have a family of normal curves. Some representations of the normal curve are displayed in the following slides. 9-52

Learning Goal 5:Properties of the Normal Dist. Here, means are different (  = 10, 15, and 20) while standard deviations are the same (  = 3) Here, means are the same (  = 15) while standard deviations are different (  = 2, 4, and 6).

9-54 Normal distributions with the same mean but with different standard deviations. Normal distributions with the same mean but with different standard deviations. Learning Goal 5: Properties of the Normal Distribution

9-55 Normal distributions with different means but with the same standard deviation. Normal distributions with different means but with the same standard deviation.

Learning Goal 5: Properties of the Normal Distribution 9-56 Normal distributions with different means and different standard deviations. Normal distributions with different means and different standard deviations.

Learning Goal 5: Describing a Normal Dist. μ located at the center of the symmetrical curve σ controls the spread The exact curve for a particular normal distribution is described by its Mean (μ) and Standard Deviation (σ). Normal Distribution Notation: N(μ,σ)

Learning Goal 5: Properties of the Normal Distribution These normal curves have similar shapes, but are located at different points along the x-axis. Also, the larger the standard deviation, the more spread out the distribution, and the curves are symmetrical about the mean. A normal distribution is a continuous, symmetrical, bell- shaped distribution. 9-58

Learning Goal 5: Properties of the Normal Distribution Summary of the Properties of the normal Distribution: The curve is continuous. The curve is bell-shaped. The curve is symmetrical about the mean. The mean, median, and mode are located at the center of the distribution and are equal to each other. The curve is unimodal (single mode) The curve never touches the x-axis. The total area under the normal curve is equal to

Learning Goal 5: Not Normal Curves Why a)Normal curve gets closer and closer to the horizontal axis, but never touches it. b)Normal curve is symmetrical. c)Normal curve has a single peak. d)Normal curve tails do not curve away from the horizontal axis.

Learning Goal 5: More Normal Distribution For a normal distribution the Mean (μ) is located at the center of the single peak and controls location of the curve on the horizontal axis. The standard deviation (σ) is located at the inflection points of the curve and controls the spread of the curve.

Learning Goal 5: Inflection Points The point on the curve where the curve changes from falling more steeply to falling less steeply (change in curvature – concave down to concave up). Located one standard deviation (σ) from the mean (μ). Allows us to visualize on any normal curve the width of one standard deviation. Inflection point

Learning Goal 5: More Normal Model There is a Normal model for every possible combination of mean and standard deviation. – We write N(μ,σ) 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 are not numerical summaries of the data. They are part of the model and a model is more like a population. They don’t come from the data. They are numbers that we choose to help specify the model. Such numbers are called parameters of the model.

Learning Goal 5: More Normal Model Summaries of data, like the sample mean and standard deviation, are written with Latin letters. Such summaries of data are called statistics. When we standardize Normal data, we still call the standardized value a z-score also, and we write

Learning Goal 5: Standardizing the Normal Distribution All normal distributions are the same general shape and share many common properties. Normal distribution notation: N(μ,σ). We can make all normal distributions the same by measuring them in units of standard deviation (σ) about the mean (μ), z-scores. This is called standardizing and gives us the Standard Normal Curve.

Learning Goal 5: Standardizing the Normal Distribution How do linear transformations apply to z- scores? When we convert a data value to a z- scores, we are shifting it by the mean (to set the scale at 0) and then rescaling by the standard deviation (to reset the standard deviation to 1). – Standardizing into z-scores does not change the shape of the distribution. – Standardizing into z-scores changes the center by making the mean 0. – Standardizing into z-scores changes the spread by making the standard deviation 1.

Learning Goal 5: Standardizing the Normal Dist. We can standardize a variable that has a normal distribution to a new variable that has the standard normal distribution using the z-score formula: Substitute your variable as x Subtract the mean from your variable Then divide by your Standard Deviation BAM! Pops out your z-score

Learning Goal 5: Standardizing the Normal Dist.

Standardizing Data into z-scores The Standard Normal Distribution

Learning Goal 5: Standardizing the Normal Dist.

Results in a Standardized Normal Distribution (curve) One Distribution → One set of areas under the curve → One Table

Learning Goal 5: Standardizing the Normal Distribution Subtracting Mu from each value X just moves the curve around, so values are centered on 0 instead of on Mu. Once the curve is centered, dividing each value by sigma>1 moves all values toward 0, smushing the curve.

Learning Goal 5: Standard Normal Dist. - Example

The area under the original curve and the standard normal curve are the same. Learning Goal 5: Standard Normal Dist. - Example

Learning Goal 5: The Standard Normal Curve Let x be a normally distributed variable with mean μ and standard deviation σ, and let a and b be real numbers with a < b. The percentage of all possible observations of x that lie between a and b is the same as the percentage of all possible observations of z that lie between (a −μ)/σ and (b−μ)/σ. This latter percentage equals the area under the standard normal curve between (a −μ)/σ and (b−μ)/σ.

Learning Goal 5: Standard Normal Curve and z-scores Same as with any Normal Distribution. A z-score gives us an indication of how unusual a value is because it tells us how far it is from the mean. A data value that sits right at the mean, has a z-score equal to 0. A z-score of 1 means the data value is 1 standard deviation above the mean. A z-score of –1 means the data value is 1 standard deviation below the mean.

Learning Goal 5: Standard Normal Curve and z-scores How far from 0 does a z-score have to be to be interesting or unusual? z-scores beyond -2 or 2 are considered unusual. Remember that a negative z-score tells us that the data value is below the mean, while a positive z-score tells us that the data value is above the mean.

Learning Goal 5: The Standard Normal Model Once we have standardized, we need only one model: – The N(0,1) model is called the Standard Normal model (or the Standard Normal distribution). Be careful—don’t use a Normal model for just any data set, since standardizing does not change the shape of the distribution.

Learning Goal 5: Properties of the Standard Normal Dist. Shape – normal curve Mean (μ) = 0 Standard Deviation (σ) = 1 Horizontal axis scale – Z score No vertical axis Notation: N(0, 1)

Learning Goal 5: Standard Normal Dist. Problem Which one of the following is a FALSE statement about the standard normal distribution? a)The mean is greater than the median. b)It is symmetric. c)It is bell-shaped. d)It has one peak.

Learning Goal 5: Standard Normal Dist. Problem If you knew that the  = 0 and  = 3, which normal curve would match the data? a)Dataset 1 b)Dataset 2

Learning Goal 5: Standard Normal Dist. Problem Which one of the following is a FALSE statement about the standard normal curve? a)Its standard deviation  can vary with different datasets. b)It is bell-shaped. c)It is symmetric around 0. d)Its mean  = 0.

Learning Goal 5: Standard Normal Dist. Problem Suppose the lengths of sport-utility vehicles (SUV) are normally distributed with mean  = 190 inches and standard deviation  = 5 inches. Marshall just bought a brand-new SUV that is inches long and he is interested in knowing what percentage of SUVs is longer than his. Using his statistical knowledge, he drew a normal curve and labeled the appropriate area of interest. Which picture best represents what Marshall drew? a)Plot A b)Plot B