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Describing Location in a Distribution: Percentiles and z-Scores

Learn how to describe the location of a value in a distribution using percentiles and z-scores. See examples and compare performances using standardized scores.

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Describing Location in a Distribution: Percentiles and z-Scores

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  1. Chapter Two 2, 6, 8, 10, 12, 14, 16, 20, 22, 32, 42, 46, 48, 50, 52, 54, 56, 58, 60, 64, 66, 68

  2. Describing Location in a Distribution • Measuring Position: Percentiles • One way to describe the location of a value in a distribution is to tell what percent of observations are less than it. Definition: The pth percentile of a distribution is the value with p percent of the observations less than it. Example, p. 85 Jenny earned a score of 86 on her test. How did she perform relative to the rest of the class? 6 7 7 2334 7 5777899 8 00123334 8 569 9 03 6 7 7 2334 7 5777899 8 00123334 8 569 9 03 Her score was greater than 21 of the 25 observations. Since 21 of the 25, or 84%, of the scores are below hers, Jenny is at the 84th percentile in the class’s test score distribution.

  3. a.) What percentile is Johnny in if he scored a 72? b.) What percentile is Jill in if she scored a 93%. • 6 7 • 7 2334 • 7 5777899 • 8 00123334 • 8 569 • 9 03 Alternate definition: Some textbooks use less than of equal to in their definition of percentile. How will this change the percentile that Jill is in?

  4. Describing Location in a Distribution • Cumulative Relative Frequency Graphs A cumulative relative frequency graph (or ogive) displays the cumulative relative frequency of each class of a frequency distribution.

  5. Interpreting Cumulative Relative Frequency Graphs Describing Location in a Distribution Use the graph from page 88 to answer the following questions. • Was Barack Obama, who was inaugurated at age 47, unusually young? • Estimate and interpret the 65th percentile of the distribution 65 11 58 47

  6. Describing Location in a Distribution • Measuring Position: z-Scores • A z-score tells us how many standard deviations from the mean an observation falls, and in what direction. Definition: If x is an observation from a distribution that has known mean and standard deviation, the standardized value of x is: A standardized value is often called a z-score. Z-scores can have negative, zero, or positive values. A z-score of zero is for values that are exactly the same as the mean. Negative z-scores are for values less than the mean and positive z-scores and for values greater than the mean. Most of the z-scores we see are between -3.49 and 3.49.

  7. Jenny earned a score of 86 on her test. The class mean is 80 and the standard deviation is 6.07. What is her standardized score?

  8. Describing Location in a Distribution • Using z-scores for Comparison We can use z-scores to compare the position of individuals in different distributions. Example, p. 91 Jenny earned a score of 86 on her statistics test. The class mean was 80 and the standard deviation was 6.07. She earned a score of 82 on her chemistry test. The chemistry scores had a fairly symmetric distribution with a mean 76 and standard deviation of 4. On which test did Jenny perform better relative to the rest of her class?

  9. The single season home run record for Major League Baseball has been set just three times since Babe Ruth hit 60 home runs in 1927. Roger Maris hit 61 in 1961, Mark McGwire hit 70 in 1998 and Barry Bonds hit 73 in 2001. In the absolute sense Barry Bonds has the best season, since he hit the most home runs. But baseball historians suggest that hitting home runs in different eras is easier than others (due to many factors). Given the following information compute the standardized scores for each player and compare to find the best performance. • Year Player HR Mean SD • 1927 B. Ruth 60 7.2 9.7 • 1961 R. Maris 61 18.8 13.4 • 1998 M McGwire 70 20.7 12.7 • 2001 B. Bonds 73 21.4 13.2

  10. B. Ruth z-score of 5.44 • R. Maris z-score of 3.16 • M. McGwire z-score of 3.87 • B. Bonds z-score of 3.91 • Babe Ruth can still claim he had the best single-season home run champ, relatively speaking.

  11. Given the following set of data. 4, 0, 1, 0, 10, 5, 0, 12, 12, 10, 10, 8, 7, 12, 0, 5, 15, 5, 12, 8, 7, 4, 15, 15 a.) Make a dotplot and comment on the distribution b.) Add 5 to all data points and make another dotplot and comment on distribution. Also, comment on how the distribution has changed from part a. c.) Multiply all the original data points by 2 and make another dotplot and comment on the distribution. Also, comment on how the distribution has changed from original data. d.) Multiply all original data by 3 and add 2 and make another dotplot and comment on distribution. Also, comment on how the distribution has changed from the original data. (At the end you should have 4 dotplots, I would suggest drawing one number line from 0 to 45 and drawing all dotplots on same number line one above the other.)

  12. Original Data M = 7.5 Range = 15 – 0 = 15 Shape = Uniform 0 15 M = 12.5 (original + 5) Range = 20 – 5 = 15 (same as above) Shape = Uniform (same as above) Original Date + 5 M = 15 (original x 2) Range = 30 – 0 = 30 (original x 2) Shape = Uniform (same as above) 0 15 Original Data x 2 0 15 M = 24.5 (original x 3 + 2) Range = 47 – 2 = 45 (original range x 3) Shape = Uniform (same as above) Original Data x 3 + 2 0 15

  13. Transforming converts the original observations from the original units of measurements to another scale. Transformations can affect the shape, center, and spread of a distribution. Describing Location in a Distribution • Transforming Data Effect of Multiplying (or Dividing) by a Constant Effect of Adding (or Subracting) a Constant • Multiplying (or dividing) each observation by the same number b (positive, negative, or zero): • multiplies (divides) measures of center and location by b • multiplies (divides) measures of spread by |b|, but • does not change the shape of the distribution • Adding the same number a (either positive, zero, or negative) to each observation: • adds a to measures of center and location (mean, median, quartiles, percentiles), but • Does not change the shape of the distribution or measures of spread (range, IQR, standard deviation).

  14. 1.) Enter the following temperatures in Celsius into the Calculator 3, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 12, 14 Press STAT Enter 1:Edit Place values into L1 2.) In L2 enter all the values that are 32 degrees larger than the ones in L1: (L1 + 32) 3.) In L3 enter all the values that are: (9/5) L1 + 32 4.) Press STAT Arrow over Calc menu and then press enter twice. Press STAT, go to Calc menu and then press enter, 2nd , #2, enter Press STAT, go to Calc menu and then press enter, 2nd, #3, enter. 5.) Make Box and Whisker plots for each set of data and comment on distribution. Use to find measures of center and spread Use to find shape and possible outliers.

  15. Describing Location in a Distribution • Density Curves • In Chapter 1, we developed a kit of graphical and numerical tools for describing distributions. Now, we’ll add one more step to the strategy. Exploring Quantitative Data • Always plot your data: make a graph. • Look for the overall pattern (shape, center, and spread) and for striking departures such as outliers. • Calculate a numerical summary to briefly describe center and spread. 4. Sometimes the overall pattern of a large number of observations is so regular that we can describe it by a smooth curve.

  16. Describing Location in a Distribution • Density Curve • Definition: • A density curve is a curve that • is always on or above the horizontal axis, and • has area exactly 1 underneath it. • A density curve describes the overall pattern of a distribution. The area under the curve and above any interval of values on the horizontal axis is the proportion of all observations that fall in that interval. Go to excel spreadsheet The overall pattern of this histogram of the scores of all 947 seventh-grade students in Gary, Indiana, on the vocabulary part of the Iowa Test of Basic Skills (ITBS) can be described by a smooth curve drawn through the tops of the bars.

  17. Distinguishing the Median and Mean of a Density Curve • The median of a density curve is the equal area point, the point that divides the area under the curve in half. • The mean of a density curve is the balance point, at which the curve would balance if made of solid material • The median and mean are the same for symmetric curves. The mean of a skewed curve is pulled away from the median in the direction of the tail.

  18. Mean and Standard Deviation of a Density Curve The density curve is an idealized description of a distribution of data, we need to distinguish between the mean and standard deviation of the density curve and the mean and standard deviation of actual observations. Actual Density Mean x µ (mu) Standard Sxσx (sigma) deviation The area under the curve for a specific range is equal to the proportion of the observation under that same range!!!

  19. Describing Location in a Distribution • Describing Density Curves • Our measures of center and spread apply to density curves as well as to actual sets of observations. Distinguishing the Median and Mean of a Density Curve The median of a density curve is the equal-areas point, the point that divides the area under the curve in half. The mean of a density curve is the balance point, at which the curve would balance if made of solid material. The median and the mean are the same for a symmetric density curve. They both lie at the center of the curve. The mean of a skewed curve is pulled away from the median in the direction of the long tail.

  20. Section 2.1Describing Location in a Distribution Summary In this section, we learned that… • There are two ways of describing an individual’s location within a distribution – the percentile and z-score. • A cumulative relative frequency graph allows us to examine location within a distribution. • It is common to transform data, especially when changing units of measurement. Transforming data can affect the shape, center, and spread of a distribution. • We can sometimes describe the overall pattern of a distribution by a density curve (an idealized description of a distribution that smooths out the irregularities in the actual data).

  21. Looking Ahead… In the next Section… • We’ll learn about one particularly important class of density curves – the Normal Distributions • We’ll learn • The 68-95-99.7 Rule • The Standard Normal Distribution • Normal Distribution Calculations, and • Assessing Normality

  22. Normal Distributions • Normal Distributions • One particularly important class of density curves are the Normal curves, which describe Normal distributions. • All Normal curves are symmetric, single-peaked, and bell-shaped • A Specific Normal curve is described by giving its mean µ and standard deviation σ. Two Normal curves, showing the mean µ and standard deviation σ.

  23. Normal Distributions • Normal Distributions • Definition: • A Normal distribution is described by a Normal density curve. Any particular Normal distribution is completely specified by two numbers: its mean µ and standard deviation σ. • The mean of a Normal distribution is the center of the symmetric Normal curve. • The standard deviation is the distance from the center to the change-of-curvature points on either side. • We abbreviate the Normal distribution with mean µ and standard deviation σ as N(µ,σ). Normal distributions are good descriptions for some distributions of real data. Normal distributions are good approximations of the results of many kinds of chance outcomes. Many statistical inference procedures are based on Normal distributions.

  24. The 68-95-99.7 Rule Normal Distributions Although there are many Normal curves, they all have properties in common. • Definition:The 68-95-99.7 Rule (“The Empirical Rule”) • In the Normal distribution with mean µ and standard deviation σ: • Approximately 68% of the observations fall within σ of µ. • Approximately 95% of the observations fall within 2σ of µ. • Approximately 99.7% of the observations fall within 3σ of µ.

  25. Example, p. 113 Normal Distributions The distribution of Iowa Test of Basic Skills (ITBS) vocabulary scores for 7th grade students in Gary, Indiana, is close to Normal. Suppose the distribution is N(6.84, 1.55). • Sketch the Normal density curve for this distribution. • What percent of ITBS vocabulary scores are less than 3.74? • What percent of the scores are between 5.29 and 9.94?

  26. Normal Distributions • The Standard Normal Distribution • All Normal distributions are the same if we measure in units of size σ from the mean µ as center. Definition: The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1. If a variable x has any Normal distribution N(µ,σ) with mean µ and standard deviation σ, then the standardized variable has the standard Normal distribution, N(0,1).

  27. Normal Distributions • The Standard Normal Table Because all Normal distributions are the same when we standardize, we can find areas under any Normal curve from a single table. • Definition:The Standard Normal Table • Table A is a table of areas under the standard Normal curve. The table entry for each value z is the area under the curve to the left of z. Suppose we want to find the proportion of observations from the standard Normal distribution that are less than 0.81. We can use Table A: P(z < 0.81) = .7910

  28. Example, p. 117 Normal Distributions • Finding Areas Under the Standard Normal Curve Find the proportion of observations from the standard Normal distribution that are between -1.25 and 0.81. Can you find the same proportion using a different approach? 1 - (0.1056+0.2090) = 1 – 0.3146 = 0.6854

  29. Normal Distributions • Normal Distribution Calculations How to Solve Problems Involving Normal Distributions • State: Express the problem in terms of the observed variable x. • Plan: Draw a picture of the distribution and shade the area of interest under the curve. • Do: Perform calculations. • Standardizex to restate the problem in terms of a standard Normal variable z. • Use Table A and the fact that the total area under the curve is 1 to find the required area under the standard Normal curve. • Conclude: Write your conclusion in the context of the problem.

  30. When Tiger Woods hits his driver, the distance the ball travels can be described by N(304, 8). What percent of Tiger’s drives travel between 305 and 325 yards? Normal Distributions • Normal Distribution Calculations Using Table A, we can find the area to the left of z=2.63 and the area to the left of z=0.13. 0.9957 – 0.5517 = 0.4440. About 44% of Tiger’s drives travel between 305 and 325 yards.

  31. Given the following test scores one class 85, 75, 98, 78, 85, 86, 95, 92, 93, 89, 84 ,86, 82, 70 • A.) Sketch a Normal distribution • (Hint: use table A for all of the following questions) • B.) What percent of the scores should fall in the B range (85 to 92)? • C.) What percent of the scores should fall in the D or C range (77 to 84)? • D.) What percent of the scores should be above an 80%? • E.) What percent of the scores should fall between an 85% and a 90%? Approximately 33% or .3263 Approximately 29% or .2855 Approximately 72% or .7228 Approximately 25% or .2458

  32. The heights of 3 year old females are approximately Normal with a mean of 94.5 cm and a standard deviation of 4 cm. What is the interquartile range? Let x be the height of a randomly selected female 3 year old, where the distribution is N(94.5, 4). Using Table A, the closest number to 75% is .7486 , which corespondes to a z-score of .67. By the symmetric property, 25% would have to be a z-score of -.67. Solve for x .67 = x – 94.5 and -.67 = x – 94.5 4 4 x = 97.18 cm x = 91.82 IQR = 97.18 – 91.82 = 5.36 cm The interquartile range of a randomly selected 3 year old female is about 5.36 cm. 82.5 86.5 90.5 94.5 98.5 102.5 106.5

  33. Normal Distributions • Assessing Normality • The Normal distributions provide good models for some distributions of real data. Many statistical inference procedures are based on the assumption that the population is approximately Normally distributed. Consequently, we need a strategy for assessing Normality. • Plot the data. • Make a dotplot, stemplot, or histogram and see if the graph is approximately symmetric and bell-shaped. • Check whether the data follow the 68-95-99.7 rule. • Count how many observations fall within one, two, and three standard deviations of the mean and check to see if these percents are close to the 68%, 95%, and 99.7% targets for a Normal distribution.

  34. Normal Distributions • Normal Probability Plots • Most software packages can construct Normal probability plots. These plots are constructed by plotting each observation in a data set against its corresponding percentile’s z-score. Interpreting Normal Probability Plots If the points on a Normal probability plot lie close to a straight line, the plot indicates that the data are Normal. Systematic deviations from a straight line indicate a non-Normal distribution. Outliers appear as points that are far away from the overall pattern of the plot.

  35. Section 2.2Normal Distributions Summary In this section, we learned that… • The Normal Distributions are described by a special family of bell-shaped, symmetric density curves called Normal curves. The mean µ and standard deviation σ completely specify a Normal distribution N(µ,σ). The mean is the center of the curve, and σ is the distance from µ to the change-of-curvature points on either side. • All Normal distributions obey the 68-95-99.7 Rule, which describes what percent of observations lie within one, two, and three standard deviations of the mean.

  36. Section 2.2Normal Distributions Summary In this section, we learned that… • All Normal distributions are the same when measurements are standardized. The standard Normal distribution has mean µ=0 and standard deviation σ=1. • Table A gives percentiles for the standard Normal curve. By standardizing, we can use Table A to determine the percentile for a given z-score or the z-score corresponding to a given percentile in any Normal distribution. • To assess Normality for a given set of data, we first observe its shape. We then check how well the data fits the 68-95-99.7 rule. We can also construct and interpret a Normal probability plot.

  37. Looking Ahead… In the next Chapter… • We’ll learn how to describe relationships between two quantitative variables • We’ll study • Scatterplots and correlation • Least-squares regression

  38. Normal Distributions • Assessing Normality • The Normal distributions provide good models for some distributions of real data. Many statistical inference procedures are based on the assumption that the population is approximately Normally distributed. Consequently, we need a strategy for assessing Normality. • Plot the data. • Make a dotplot, stemplot, or histogram and see if the graph is approximately symmetric and bell-shaped. • Check whether the data follow the 68-95-99.7 rule. • Count how many observations fall within one, two, and three standard deviations of the mean and check to see if these percents are close to the 68%, 95%, and 99.7% targets for a Normal distribution.

  39. Normal Distributions • Normal Probability Plots • Most software packages can construct Normal probability plots. These plots are constructed by plotting each observation in a data set against its corresponding percentile’s z-score. Interpreting Normal Probability Plots If the points on a Normal probability plot lie close to a straight line, the plot indicates that the data are Normal. Systematic deviations from a straight line indicate a non-Normal distribution. Outliers appear as points that are far away from the overall pattern of the plot.

  40. Section 2.2Normal Distributions Summary In this section, we learned that… • The Normal Distributions are described by a special family of bell-shaped, symmetric density curves called Normal curves. The mean µ and standard deviation σ completely specify a Normal distribution N(µ,σ). The mean is the center of the curve, and σ is the distance from µ to the change-of-curvature points on either side. • All Normal distributions obey the 68-95-99.7 Rule, which describes what percent of observations lie within one, two, and three standard deviations of the mean.

  41. Section 2.2Normal Distributions Summary In this section, we learned that… • All Normal distributions are the same when measurements are standardized. The standard Normal distribution has mean µ=0 and standard deviation σ=1. • Table A gives percentiles for the standard Normal curve. By standardizing, we can use Table A to determine the percentile for a given z-score or the z-score corresponding to a given percentile in any Normal distribution. • To assess Normality for a given set of data, we first observe its shape. We then check how well the data fits the 68-95-99.7 rule. We can also construct and interpret a Normal probability plot.

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