3.3

𝒁-SCORE Who is taller, a man 73 inches tall or a woman 68 inches tall? The obvious answer is that the man is taller. However, men are taller than women on the average. Suppose the question is asked this way: Who is taller relative to their gender, a man 73 inches tall or a woman 68 inches tall?

One way to answer this question is with a 𝑧-score.

tricks IQR formula, lower/upper outlier

(IQR)=Q3-Q1 Lower Outlier Boundary = 𝑄1 - 1.5(IQR) Upper Outlier Boundary = 𝑄3 + 1.5(IQR

One method for detecting outliers involves a measure called the Interquartile Range. INTERQUARTILE RANGE The interquartile range is found by subtracting the __________ quartile from the __________ quartile. IQR = __________________________

(IQR)=Q3-Q1 Lower Outlier Boundary = 𝑄1 - 1.5(IQR) Upper Outlier Boundary = 𝑄3 + 1.5(IQR) any value greater than these is an outlier

The 𝑧-score of an individual data value tells how many _____________________________ that value is from its population mean. For example, a value one standard deviation above the mean has a 𝑧-score of ____________ and a value two standard deviations below the mean has a 𝑧-score of ____________.

standard deviations, z=1,z=-2

trick finding a number from a z score formula

u+z(a) (z=zscore)

The temperature in a downtown location is measured for eight consecutive days during the summer. The readings, in Fahrenheit, are 81.2 85.6 89.3 91.0 83.2 8.45 79.5 87.8 Which reading is an outlier? Is the outlier an error or is it possible that it is correct?

8.45, an error

FIVE-NUMBER SUMMARY FOR A DATA SET The ___________________________________________ of a data set consists of the median, the first quartile, the third quartile, the smallest value, and the largest value. These values are generally arranged in order.

5 number summary

The 1-Var Stats command in the TI-84 PLUS Calculator displays a list of the most common parameters and statistics for a given data set. This command is accessed by pressing STAT and then highlighting the CALC menu. EXAMPLE: QUARTILES ON THE TI-84 PLUS Step 1: Enter the data into L1. Step 2: Press STAT and highlight the CALC menu. Step 3: Run the 1-Var Stats command

5 number summary (Min, Q1, Median,Q3, Max) given by the 1-Var stats command

QR METHOD FOR DETECTING OUTLIERS The most frequent method used to detect outliers in a data set is the IQR Method. The procedure for the IQR Method is: Step 1: Find the first quartile 𝑄1, and the third quartile 𝑄3. Step 2: Compute the interquartile range: IQR = 𝑄3 − 𝑄1. Step 3: Compute the outlier boundaries. These boundaries are the cutoff points for determining outliers: Lower Outlier Boundary = 𝑄1 - 1.5(IQR) Upper Outlier Boundary = 𝑄3 + 1.5(IQR) Step 4: Any data value that is less than the lower outlier boundary or greater than the upper outlier boundary is considered to be an outlier. EXAMPLE: The following table presents the number of students absent in a middle school in northwestern Montana for each school day in January. Identify any outliers. 65 67 71 57 51 49 44 41 59 49 42 56 45 77 44 42 45 46 100 59 53 51

59-45=14 (lower)Q1-1.5(IQR)= 45-1.5(14)=24 (higher)Q3-1.5(14)=80 therefore since 100 is greater than 80 it is an outlier

A National Center for Health Statistics study states that the mean height for adult men in the U.S. is 𝜇 = 69.4 inches, with a standard deviation of 𝜎 = 3.1 inches. The mean height for adult women is 𝜇 = 63.8 inches, with a standard deviation of 𝜎 = 2.8 inches. Who is taller relative to their gender, a man 73 inches tall, or a woman 68 inches tall?

73-69.4/3.1=1.16 68-63.8/2.8=1.50 (woman is taller)

QUARTILES In a previous section, we learned how to compute the mean and median of a data set as measures of the center. Sometimes, it is useful to compute measures of position other than the center to get a more detailed description of the distribution. Quartiles divide a data set into four approximately equal pieces. QUARTILES: Every data set has three quartiles: The __________________________, denoted 𝑄1 separates the lowest __________ of the data from the highest __________. The __________________________, denoted 𝑄2 separates the lowest __________ of the data from the highest __________. 𝑄2 is the same as the median. The __________________________, denoted 𝑄3 separates the lowest __________ of the data from the highest __________.

First quartile, 25%-75%, second quartile, 50%-50%, Q2=median, Third quartile,75%-25%

OBJECTIVE 3 COMPUTE THE PERCENTILES OF A DATA SET Quartiles describe the shape of a distribution by dividing it into fourths. Sometimes it is useful to divide a data set into a greater number of pieces to get a more detailed description of the distribution. ________________________ divide a data set into hundredths.

Percentiles

EXAMPLE: The following table presents the annual rainfall, in inches, in Los Angeles during the month of February over several years. Compute the quartiles for the data. 0.00 0.08 0.13 0.14 0.16 0.17 0.20 0.29 0.56 0.67 0.70 0.92 1.22 1.30 1.48 1.64 1.72 1.90 2.37 2.58 2.84 3.06 3.12 3.21 3.29 3.54 3.57 3.71 4.13 4.27 4.37 4.64 4.89 4.94 5.54 6.10 6.61 7.89 7.96 8.03 8.87 8.91 11.02 12.75 13.68

SOLUTION: L=0.25(45)=11.25 (round to 12) the 12th position of the first quartile is 0.83 For the second quartile we compute the median= 3.12 For the third quartile L= 0.75(45)=33.75 (round up to 34) the 34th position in the third quartile is 4.89

Recall the Los Angeles annual rainfall data. Compute the five-number summary. When using the TI-84 PLUS Calculator, the five-number summary is given by the 1-Var Stats command. 0.00 0.08 0.13 0.14 0.16 0.17 0.20 0.29 0.56 0.67 0.70 0.92 1.22 1.30 1.48 1.64 1.72 1.90 2.37 2.58 2.84 3.06 3.12 3.21 3.29 3.54 3.57 3.71 4.13 4.27 4.37 4.64 4.89 4.94 5.54 6.10 6.61 7.89 7.96 8.03 8.87 8.91 11.02 12.75 13.68

Solution:

Let 𝑥 be a value from a population with mean 𝜇 and standard deviation 𝜎. The z-score for 𝑥 is 𝑧 =

The z-score for 𝑥 is 𝑧 = x-u ------- a

OBJECTIVE 6 CONSTRUCT BOXPLOTS TO VISUALIZE THE FIVE-NUMBER SUMMARY AND OUTLIERS A _____________________ is a graph that presents the five-number summary along with some additional information about a data set. There are several different kinds of boxplots. The one we describe here is sometimes called a __________________________________.

boxplot, modified boxplot

Trick to get a 5 number summary

file info in ascending order on a list, hit vars-1 (Min,Q1,med,Q3,Max)

EXAMPLE: The following table presents the annual rainfall in Los Angeles during the month of February over several years. One year, the rainfall was 1.90. What percentile does this correspond to? SOLUTION: 0.00 0.08 0.13 0.14 0.16 0.17 0.20 0.29 0.56 0.67 0.70 0.92 1.22 1.30 1.48 1.64 1.72 1.90 2.37 2.58 2.84 3.06 3.12 3.21 3.29 3.54 3.57 3.71 4.13 4.27 4.37 4.64 4.89 4.94 5.54 6.10 6.61 7.89 7.96 8.03 8.87 8.91 11.02 12.75 13.68

n=45, for the 60th percentile we compute L=(60/100)45=27 the average numbers in the 27th and 28th positions. 60th percentile= 3.57+3.58 ------------=3.575 2

An _______________ is a value that is considerably larger or considerably smaller than most of the values in a data set. Some outliers result from errors; for example a misplaced decimal point may cause a number to be much larger or smaller than the other values in a data set. Some outliers are correct values, and simply reflect the fact that the population contains some extreme values.

outlier

Computing percentile Step 1: Arrange the data in increasing order. Step 2: Let n be the number of values in the data set. For the pth percentile, compute the value p L= ------ * n 100 Step 3: If L is a whole number, then the pth percentile is the average of the number in position L and the number in position L+1 If L is not a whole number, round it up to the next higher whole number. The pth percentile is the number in the position corresponding to the rounded-up value.

p L= ------ * n 100 L, L+1 percentile 1 value +p2 ---------------------------- 2

tricks z- score formula

x-u ----- a