Describing Location in a Distribution

Describing Location in a Distribution
Chapter 2 Notes Describing Location in a Distribution

Where do I stand compared to others?
2.1 – Measures of Relative Standing and Density Curves Where do I stand compared to others?

NOTATION: Sample Population Mean x  “mu” Standard Deviation Sx or s  “sigma” Note: Never use x on the calc

Sally wants to go to college
Sally wants to go to college. She recently took a really hard AP Stats test and got her score back. She went to several of her friends to see their scores.

Chapter 1 Test Results 4 5 6 7 8 9 8 9 4 0 9 0 0

Standardizing Values and Z-Scores
We are going to see how the test score looks based on the variability of the data using the standard deviation based on the center of our data, the mean. This tells us how many standard deviations the test score is away from the mean.

Standardizing Values and Z-Scores
Standardized values have no units. 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 below the mean, while a positive z-score tells us that the data value is above the mean.

Benefits of Standardizing
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.

Back to z-scores Standardizing data into z-scores shifts the data by subtracting the mean and rescales the values by dividing by their standard deviation. 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.

To standardize: Convert the mean to 0. Convert the standard deviation to 1.

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 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.

Example #1 Find Sally’s z-score and comment on how she did on the test. She is barely (only 0.36 standard deviations) above the average test score. She is slightly above average.

x –   266 – 266 16 0 . 16 Z = = = = That’s the mean! Example #2
The distribution of the duration of human pregnancies (i.e. the number of days between conception and birth) has been found with mean  = 266 and the  = 16. a. What is the Z-Score for a human pregnancy of 266 days? x –   266 – 266 16 0 . 16 Z = = = = That’s the mean!

One standard deviation below the mean
Example #2 The distribution of the duration of human pregnancies (i.e. the number of days between conception and birth) has been found to be approximately normal with mean  = 266 and the  = 16. b. What is the Z-Score for a human pregnancy of 250 days? x –   250 – 266 16 -16 16 Z = = = = -1 One standard deviation below the mean

Dalmatian x –   45 – 50 3.3 -5 . 3.3 Z = = = = -1.515 Boxer x –  
Example #3 Adult female Dalmations weigh an average of 50 pounds with a standard deviation of 3.3 pounds. Adult female Boxers weigh an average of 57.5 pounds with a standard deviation of 1.7 pounds. Mike owns an underweight Dalmatian and an underweight Boxer. The Dalmatian weighs 45 pounds and the Boxer weighs 52 pounds. Which dog is more underweight? Explain. Dalmatian x –   45 – 50 3.3 -5 . 3.3 Z = = = = -1.515 Boxer x –   52 – 57.5 1.7 -5.5 . 1.7 Z = = = = -3.235 The Boxer’s weight is VERY low comparative to other Boxers.

Hmm…so how unusual is ?

Percentile: Percent of the observations at or below a value

Chebshev’s Inequality
In any distribution, the percent of observations falling within k standard deviations of the mean is at least -sd sd mean

Example #4 Find the percentile ranking of the following: One standard deviation from the mean Two standard deviations from the mean Three standard deviations from the mean Four standard deviations from the mean

Density Curve: Mathematical model that describes a set of data. Above x-axis Total area under curve = 1

a. Verify that this is a density curve.
Example #5 Using the following uniform density curve, answer the question: a. Verify that this is a density curve. A = bh A = (8)(0.125) A = 1

Example #5 Using the following uniform density curve, answer the question: b. What is the probability that the random variable has a value less than 3? A = bh A = (3)(0.125) A = 0.375

Example #5 Using the following uniform density curve, answer the question: c. What is the probability that the random variable has a value between 3 and 5? A = bh A = (2)(0.125) A = 0.25

d. What is the percentile for the variable that has a value of 6?
Example #5 Using the following uniform density curve, answer the question: d. What is the percentile for the variable that has a value of 6? A = bh A = (6)(0.125) A = 0.75

e. What value for the variable is in the 25th percentile?
Example #5 Using the following uniform density curve, answer the question: e. What value for the variable is in the 25th percentile? A = bh 0.25 = (b)(0.125) 2 = b

In a density curve: The mean, median, and quartiles can be located by eye.

The mean, , is the balance point of the curve, if it were made of solid material.

The median, M, divides the area under the curve in half.
The quartiles with the median divide the curve into quarters.

The mean and the median are the same only if the distribution is symmetrical. The median is a measure of center that is resistant to skew and outliers. The mean is not. Mean Median

Mean Median Mean Median

Example #6 A group of 78 third-grade students in a Midwestern elementary school took a “self-concept” test that measured how well they felt about themselves. Higher scores indicate more positive self-concepts. A histogram for these students’ self-concept scores are given below. Draw an appropriate density curve for summarizing the histogram on the graph above. How would you describe the shape of this density curve?

Example #6 A group of 78 third-grade students in a Midwestern elementary school took a “self-concept” test that measured how well they felt about themselves. Higher scores indicate more positive self-concepts. A histogram for these students’ self-concept scores are given below. Draw an appropriate density curve for summarizing the histogram on the graph above. How would you describe the shape of this density curve? Skew Left

A = mode A = A = mean B = median B = n/a B = median C = mean C = n/a
Example #7 Label A, B, and C as either the mean, median, or mode for each picture. A = mode A = A = mean Mean, median, mode B = median B = n/a B = median C = mean C = n/a C = mode

a. The mean and median are equal.
Example #8 For the density curve below, which of the following is true? a. The mean and median are equal. b. The mean is greater than the median. c. The mean is less than the median d. The mean could be either greater than or less than the median e. The mean is 0.5

2.2 – Normal Distributions
Think of how stupid the average person is, and realize half of them are stupider than that.

Normal Curves: Density curve that is symmetric, unimodal, and bell-shaped It is used to compare information from different populations or to find the percentile of a certain value.

N(,) where  = mean and  = standard deviation of the population.
The mean is the center and the standard deviation is the distance from the mean.

The NORMAL(or BELL-SHAPED) DISTRIBUTION describes many different data sets, like:
Scores on a midterm exam Weights of people Calorie consumption per day Lengths of pregnancies Heights of people IQ The # of M&M’s in a 1lb bag

The 68-95-99.7 Rule (Empirical Rule):
In the normal distribution with mean  and standard deviation : 68% of the observations fall within  of the mean , or  95% of the observations fall within 2 of , or 2 99.7% of the observations fall within 3 of , or 3

Example #5 Find the area between each of the deviations.

Why are these not percentiles?

So, how do you know if it is normal?

Assessing Normality: Looks can be deceiving! Method #1: Make a histogram 1. Find the mean and sd 2. Measure the intervals for 1, 2, and 3 sd 3. Count how many observations fall between these standard deviations 4. Compare to the Rule

Remember the Dalmatians?
Example #9: Here is a sample of 25 dalmatians from a local vets office. Determine if their weights follow a normal distribution. Mean = 49.92lbs S.D. = 3.34lbs

100% 60% 5 6 9 4 39.9 43.24 46.58 49.92 53.26 56.60 59.94 Gosh, tough call!

Method #2: Normal Probability Plot 1. Plug data into a list 2. Make a Normal Probability Plot 3. If the data is normal then it will make a straight line Zscore data

Normal

Left Skew

Right Skew

Bimodal

Calculator Tip: Normal Probability Plot Statplot – Normal Probability Plot

Back to the dogs… Example #10: Make a normal probability plot of the weights of dalmatians.

Example #11 The plot shown at the right is a Normal probability plot for a set of data. The data value is plotted on the x axis, and the standardized value is plotted on the y axis. Which statement is true for these data? (a) The data are clearly Normally distributed. (b) The data are approximately Normally distributed. (c) The data are clearly skewed to the left. (d) The data are clearly skewed to the right. (e) There is insufficient information to determine the shape of the distribution.

Day 2 of 2.2!

The Standard Normal Curve:
The requirement is that the data must me normally distributed. We will use TABLE A. (In the text, you have TABLE A in the front 2 pages of the book.)

How can you find me?

Example #12 Let’s us TABLE A to find the following: Draw the picture first, shade the region you want and look up the X in Table A to find the proportion, probability (percentage) to the left of that z-score. The proportion is also known as probability that the value of a particular member of a population will fall in the given interval. a. P (Z < -2.20) =  = 1 =0

Example #12 Let’s us TABLE A to find the following: Draw the picture first, shade the region you want and look up the X in Table A to find the proportion, probability (percentage) to the left of that z-score. The proportion is also known as probability that the value of a particular member of a population will fall in the given interval. a. P (Z < -2.20) =  = 1 Z -2.20 =0

Example #12 Let’s us TABLE A to find the following: Draw the picture first, shade the region you want and look up the X in Table A to find the proportion, probability (percentage) to the left of that z-score. The proportion is also known as probability that the value of a particular member of a population will fall in the given interval. a. P (Z < -2.20) = 0.0139  = 1 Z -2.20 =0

Example #12 Let’s us TABLE A to find the following: Draw the picture first, shade the region you want and look up the X in Table A to find the proportion, probability (percentage) to the left of that z-score. The proportion is also known as probability that the value of a particular member of a population will fall in the given interval. b. P (Z > 2.20) =  = 1  = 0 Z 2.20

1 – P(Z < 2.20) = b. P (Z > 2.20) = Example #12
Let’s us TABLE A to find the following: Draw the picture first, shade the region you want and look up the X in Table A to find the proportion, probability (percentage) to the left of that z-score. The proportion is also known as probability that the value of a particular member of a population will fall in the given interval. b. P (Z > 2.20) = 1 – P(Z < 2.20) =  = 1  = 0 Z 2.20

b. P (Z > 2.20) = 1 – P(Z < 2.20) = 1 – 0.9861 = 0.0139
Example #12 Let’s us TABLE A to find the following: Draw the picture first, shade the region you want and look up the X in Table A to find the proportion, probability (percentage) to the left of that z-score. The proportion is also known as probability that the value of a particular member of a population will fall in the given interval. b. P (Z > 2.20) = 1 – P(Z < 2.20) = 1 – = 0.0139  = 1  = 0 Z 2.20

c. P(Z > -0.95) = 1 – P(Z <-0.95) = Example #12
Let’s us TABLE A to find the following: Draw the picture first, shade the region you want and look up the X in Table A to find the proportion, probability (percentage) to the left of that z-score. The proportion is also known as probability that the value of a particular member of a population will fall in the given interval. c. P(Z > -0.95) = 1 – P(Z <-0.95) =  = 1  = 0 Z -0.95

c. P(Z > -0.95) = 1 – P(Z <-0.95) = 1 – 0.1711 = 0.8289
Example #12 Let’s us TABLE A to find the following: Draw the picture first, shade the region you want and look up the X in Table A to find the proportion, probability (percentage) to the left of that z-score. The proportion is also known as probability that the value of a particular member of a population will fall in the given interval. c. P(Z > -0.95) = 1 – P(Z <-0.95) = 1 – = 0.8289  = 1  = 0 Z -0.95

Example #12 Let’s us TABLE A to find the following: Draw the picture first, shade the region you want and look up the X in Table A to find the proportion, probability (percentage) to the left of that z-score. The proportion is also known as probability that the value of a particular member of a population will fall in the given interval. d. P (Z < 1.25) =  = 1  = 0 Z 1.25

Example #12 Let’s us TABLE A to find the following: Draw the picture first, shade the region you want and look up the X in Table A to find the proportion, probability (percentage) to the left of that z-score. The proportion is also known as probability that the value of a particular member of a population will fall in the given interval. d. P (Z < 1.25) = 0.8944  = 1  = 0 Z 1.25

e. P (-1.04 < Z < 3.01) = P(Z < 3.01) – P(Z < –1.04)
Example #12 Let’s us TABLE A to find the following: Draw the picture first, shade the region you want and look up the X in Table A to find the proportion, probability (percentage) to the left of that z-score. The proportion is also known as probability that the value of a particular member of a population will fall in the given interval. e. P (-1.04 < Z < 3.01) = P(Z < 3.01) – P(Z < –1.04)  = 1 Z -1.04  = 0 Z 3.01

e. P (-1.04 < Z < 3.01) = P(Z < 3.01) – P(Z < –1.04) = – =  = 1 Z -1.04  = 0 Z 3.01

f. P (0.15 < Z < 1.41) = P(Z < 1.41) – P(Z<0.15)
Example #12 Let’s us TABLE A to find the following: Draw the picture first, shade the region you want and look up the X in Table A to find the proportion, probability (percentage) to the left of that z-score. The proportion is also known as probability that the value of a particular member of a population will fall in the given interval. f. P (0.15 < Z < 1.41) = P(Z < 1.41) – P(Z<0.15)  = 1  = 0 Z 1.41 Z 0.15

f. P (0.15 < Z < 1.41) = P(Z < 1.41) – P(Z<0.15)
= – =  = 1  = 0 Z 1.41 Z 0.15

Calculator Tip: Catalog Help Apps – CtlgHelp – Enter – Enter

Calculator Tip: Z-score Probabilities 2nd Dist – Normalcdf, (lowerbound, upperbound, mean, sd)

x –   29 – 34 3.5 -5 3.5 Z = = = = -1.4285 Example #13
Suppose that the fuel efficiency (in miles per gallon) of a Beetle varies with each tank of gas according to a normal distribution with  = 34 and standard deviation  = 3.5 miles per gallon. a. What proportion of all tanks would get 29 miles per gallon or less? x –   29 – 34 3.5 -5 3.5 Z = = = =

P (Z < ) =  = 3.5  = 1 =0 x 29 =34 Z

P (Z < ) = 0.0764  = 3.5  = 1 =0 x 29 =34 Z

Example #13 Suppose that the fuel efficiency (in miles per gallon) of a Beetle varies with each tank of gas according to a normal distribution with  = 34 and standard deviation  = 3.5 miles per gallon. b. What proportion of all tanks would get 40 miles per gallon or more? x –   40 – 34 3.5 6 . 3.5 Z = = = = 1.7143

P (Z > ) =  = 3.5  = 1 =0 =34 x 40 Z 1.71

P (Z > ) = 0.0436  = 3.5  = 1 =0 =34 x 40 Z 1.71

Example #13 Suppose that the fuel efficiency (in miles per gallon) of a Beetle varies with each tank of gas according to a normal distribution with  = 34 and standard deviation  = 3.5 miles per gallon. c. What proportion of all tanks would get between 27 and 42 miles per gallon? x –   27 – 34 3.5 -7 3.5 Z = = = = -2 x –   42 – 34 3.5 8 . 3.5 Z = = = = 2.285

P (-2 < Z < 2.29) = P(Z < 2.29) – P(Z < -2)  = 3.5  = 1
27  = 34 Z -2  = 0 Z 42 Z 2.29

P (-2 < Z < 2.29) = P(Z < 2.29) – P(Z < -2)
= – =  = 3.5  = 1 Z 27  = 34 Z -2  = 0 Z 42 Z 2.29

Example #13 Suppose that the fuel efficiency (in miles per gallon) of a Beetle varies with each tank of gas according to a normal distribution with  = 34 and standard deviation  = 3.5 miles per gallon. d. What proportion of all tanks would get 47 miles per gallon or more? Less than 47 miles per gallon? x –   47 – 34 3.5 13 3.5 Z = = = = 3.715

P (Z < 3.715) = P (Z > 3.715) =  = 1  = 1  = 0 =0 Z Z 3.72

P (Z < 3.715) = 1 P (Z > 3.715) =  = 1  = 1  = 0 =0 Z Z 3.72
 = 1  = 1  = 0 =0 Z 3.72 Z 3.72

Example #14: Golf courses have a wide range of difficulty
Example #14: Golf courses have a wide range of difficulty. Similarly, players differ in ability. In order to adjust for variations between players, they are often assigned a handicap score. To adjust for variations between courses, a handicapper decides to compare the golfer’s score against the data from the course. Suppose course A plays at a mean score of 76 with a standard deviation of 8 strokes with a normal distribution of scores. The mean score for course B is 80 with a standard deviation of 6 strokes and the scores are normally distributed. If a golfer regularly shoots an 80 on course A, what should be the comparable score on course B?

 = 8  = 76 x 80  = 6  = 80 x=?

x –   80 – 76 8 4 8 Z = = = = 0.5 x –   Z = x – 80 6 0.5 = 3 = x – 80 83 = x

Inverse Normal Probability Calculations: Un-standardizing
What if I know the percentile, but not X? Inverse Normal Probability Calculations: Un-standardizing

Example #15: What value(s) of Z cut off the region described?
a. The lowest 11%  = 1 0.11 Z=? =0

a. The lowest 11% Z = -1.23  = 1 0.11 Z=? =0

b. The highest 30%  = 1 0.30 =0 Z = ?

b. The highest 30% Z = 0.52  = 1 0.30 =0 Z = ?

c. The highest 7%  = 1 0.07 =0 Z = ?

c. The highest 7% Z = 1.48  = 1 0.07 =0 Z = ?

d. The middle 50% 75% – 25%  = 1 Z=?  = 0 Z=?

75% – 25% Z = 0.67 and Z = -0.67 d. The middle 50%
Example #15: What value(s) of Z cut off the region described? d. The middle 50% 75% – 25% Z = 0.67 and Z = -0.67  = 1 Z=?  = 0 Z=?

Want Z? Calculator Tip: Z-score given probabilities 2nd Dist – invNorm(%)

Want X? Calculator Tip: Z-score given probabilities 2nd Dist – invNorm(%, , )

Steps for the Inverse Probability Calculation:
Draw a picture Identify the z-value from the given value of the proportion – look up the proportion in the MIDDLE of Table A. Solve for x: x –   Z = x = Z() + 

Example # 16 A British company called Molebegon removes unwanted moles from gardens. In 1995, the European Union announced that the tiny moles are just too difficult to catch. They will not attempt to catch the smallest 10%. Molebegon’s past records indicate that weights of moles are normally distributed with a mean of 150 grams and a standard deviation of 32 grams. What is the cut off weight for the moles they will catch? x = Z() +   = 32 0.10 Z=? =150

x = Z() +  x = (-1.28)(32) + 150 0.10 x = -40.96 + 150 x =
Example # 16 A British company called Molebegon removes unwanted moles from gardens. In 1995, the European Union announced that the tiny moles are just too difficult to catch. They will not attempt to catch the smallest 10%. Molebegon’s past records indicate that weights of moles are normally distributed with a mean of 150 grams and a standard deviation of 32 grams. What is the cut off weight for the moles they will catch? x = Z() +   = 32 x = (-1.28)(32) + 150 0.10 x = x = grams Z=? =150

Example #17 ACT versus SAT, I There are two major tests of readiness for college, the ACT and the SAT. ACT scores are reported on a scale from 1 to 36. The distribution of ACT scores in recent years has been roughly Normal with mean = 20.9 and standard deviation = 4.8. SAT scores (prior to 2005) were reported on a scale from 400 to SAT scores have been roughly Normal with mean = 1026 and standard deviation = The following exercises are based on this information. a. Jose scores 1287 on the SAT. Assuming that both tests measure the same thing, what score on the ACT is equivalent to Jose's SAT score? Explain.

SAT  = 209  = 1026 x 1287 ACT  = 4.8  = 20.9 x=?

SAT x –   1287 – 1026 209 261 209 Z = = = = 1.25 x –   ACT Z = x – 20.9 4.8 1.25 = 5.99 = x – 20.9 26.89 = x

b. Reports on a student's ACT or SAT usually give the percentile as well as the actual score. Tonya scores 1318 on the SAT. What is her percentile? Show your method. x –   Z = 1318 – 1026 209 = = 1.40 P(Z < 1.40) =  = 209 =1026 x = 1318

b. Reports on a student's ACT or SAT usually give the percentile as well as the actual score. Tonya scores 1318 on the SAT. What is her percentile? Show your method. x –   Z = 1318 – 1026 209 = = 1.40 P(Z < 1.40) = 0.9192  = 209 The 91st percentile =1026 x = 1318

c. The quartiles of any distribution are the values with cumulative proportions 0.25 and What are the quartiles of the distribution of ACT scores? Show your method. x = Z() +   = 4.8 25% Z?  = 20.9

x = Z() +  x = (-0.67)(4.8) + 20.9 x = -3.216 + 20.9 x = 17.684 25%
c. The quartiles of any distribution are the values with cumulative proportions 0.25 and What are the quartiles of the distribution of ACT scores? Show your method. x = Z() +  x = (-0.67)(4.8) x = x = 17.684  = 4.8 25% Z?  = 20.9

c. The quartiles of any distribution are the values with cumulative proportions 0.25 and What are the quartiles of the distribution of ACT scores? Show your method. x = Z() +   = 4.8 25%  = 20.9 Z?

x = Z() +  x = (0.67)(4.8) + 20.9 x = 3.216 + 20.9 x = 24.116 25%
c. The quartiles of any distribution are the values with cumulative proportions 0.25 and What are the quartiles of the distribution of ACT scores? Show your method. x = Z() +  x = (0.67)(4.8) x = x = 24.116  = 4.8 25%  = 20.9 x=?

Describing Location in a Distribution

Similar presentations

Presentation on theme: "Describing Location in a Distribution"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Describing Location in a Distribution

Similar presentations

Presentation on theme: "Describing Location in a Distribution"— Presentation transcript:

Similar presentations

About project

Feedback