Warm Up Solve for x 2) 2x + 80 The product of a number and 9 increased by 4 is 58. Find the number. 5(x + 2) 3x 5(x + 2) + 2x + 80 + 3x = 180 9x + 4 = 58 9x = 54 5x + 10 + 2x + 80 + 3x = 180 x = 6 10x + 10 + 80 = 180 10x + 90 = 180 3) 9 – 4(2p – 1) = 45 10x = 90 4) 9x = 3(x – 2) 9 – 8p + 4 = 45 x = 9 9x = 3x - 6 13 – 8p = 45 6x = -6 -8p = 32 x = -1 p = -4

Chapter 6

Measures of Center Mean, Median, Mode and Range Chapter 6.6 Measures of Center Mean, Median, Mode and Range

Vocabulary Mean or Average Median Mode Range The sum of all the numbers divided by the total number of numbers Median The middle number when the numbers are written in order If there are two middle numbers you find the average of the two numbers Mode The number that occurs most often You can have more then one mode Range The largest number subtracted by the smallest number

Find the mean, median, mode and range 4, 2, 10, 6, 10, 7, 10 First write the numbers in order from least to greatest 2, 4, 6, 7, 10, 10, 10 Mean: 2, 4, 6, 7, 10, 10, 10 Median: 10 Mode: 10 – 2 = 8 Range:

Find the mean, median, mode and range 5, 3, 10, 13, 8, 18, 5, 17, 2, 7, 9, 10, 4, 1 First write the numbers in order from least to greatest 1, 2, 3, 4, 5, 5, 7, 8, 9, 10, 10, 13, 17, 18 Mean: Median: 1, 2, 3, 4, 5, 5, 7, 8, 9, 10, 10, 13, 17, 18 5 and 10 Mode: 18 – 1 = 17 Range:

What do you know using mean, median, mode and range? Which is greater? Which is smaller? Are any equal? 8, 5, 6, 5, 6, 6 Mean = 6 Mean, Median and Mode are all equal Median = 6 Mode = 6 Range is the smallest Range = 3

What do you know using mean, median, mode and range? Which is greater? Which is smaller? Are any equal? 161, 146, 158, 150, 156, 150, 146, 150, 150, 156, 158, 161 Mean = 153.5 Mean, is the greatest Median = 153 Range is the smallest Mean > Median Mode = 150 Median > Mode Range = 15 Mode < Mean

Calculating and Interpreting How does an outlier affect the measures of center and range? Test grades: 50, 70, 62, 80, 70, 76 Test grades: 50, 70, 62, 80, 70, 76, 100 Mean: 68 Mean: 72.6 Median: Median: 70 70 Mode: Mode: 70 70 Range: 30 Range: 50

Calculating and Interpreting How does an outlier affect the measures of center and range? 8, 5, 6, 5, 6, 6 8, 5, 6, 5, 6, 6, 15 Mean: 6 Mean: 7.3 Median: Median: 6 6 Mode: Mode: 6 6 Range: 3 Range: 10

Calculating and Interpreting How does an outlier affect the measures of center and range? 161, 146, 158, 150, 156, 150 161, 146, 158, 150, 156, 150, 200 Mean: 153.5 Mean: 160.1 Median: Median: 153 156 Mode: Mode: 150 150 Range: 15 Range: 54

Homework: Page 371 problems 18-24 even Find the mean, median, mode and range 18) 1, 2, 1, 2, 1, 3, 3, 4, 3, 20) 4, 4, 4, 4, 4, 4 22) 12, 5, 6, 15, 12, 9, 13, 1, 4, 6, 8, 14, 12 24) 161, 146, 158, 150, 156, 150

Warm Up Add up all the numbers then divide by how many numbers there are How do you find the mean? How do you find the median? How do you find the mode? How do you find the range? Find the mean, median, mode and range. 4, 2, 10, 6, 10, 7, 10 Put the numbers in order and find the middle number Find the number that occurs most often Subtract the largest number and the smallest number Mean = 7 Median = 7 Mode = 10 Range = 8

Homework: Page 371 problems 18-24 even Find the mean, median, mode and range 18) 1, 2, 1, 2, 1, 3, 3, 4, 3, 20) 4, 4, 4, 4, 4, 4 22) 12, 5, 6, 15, 12, 9, 13, 1, 4, 6, 8, 14, 12 24) 161, 146, 158, 150, 156, 150 Mean = 20/9 Median = 2 Mode = 1 and 3 Range = 3 Mean = 4 Median = 4 Mode = 4 Range = 0 Mean = 9 Median = 9 Mode = 12 Range = 14 Mean = 153.5 Median = 153 Mode = 150 Range = 15

Chapter 6.6 Stem-and-Leaf Plot

Stem-and-Leaf Plot Arrangement of digits that is used to display and order numerical data

Making a Stem-and-Leaf Plot 60, 74, 75, 63, 78, 70, 50, 74, 52, 74, 65, 78, 54 5 6 7 2 4 3 5 4 4 4 5 8 8

Practice on your own 4, 31, 22, 37, 39, 24, 2, 28, 1, 26, 28, 30, 28, 3, 20, 20, 5 1 2 3 1 2 3 4 5 0 0 2 4 6 8 8 8 0 1 7 9

Box-and-Whisker Plots Chapter 6.7 Box-and-Whisker Plots

Box-and-Whisker Plots Divides a set of data into four parts Median or Second Quartile Separates the set into two halves Numbers below the median Numbers above the median First Quartile Median of the lower half Third Quartile Median of the upper half

12, 5, 3, 8, 10, 7, 6, 5 Find the first, second and third quartiles 3, 5, 5, 6, 7, 8, 10, 12 Second = 7, 8, 10, 12 3, 5, 5, 6 First = Third = 2 3 4 5 6 7 8 9 10 11 12 13

1, 12, 6, 5, 4, 7, 5, 10, 3, 4 Find the first, second and third quartiles 1, 3, 4, 4, 5, 5, 6, 7, 10, 12 Second = 5, 6, 7, 10, 12 1, 3, 4, 4, 5 First = Third = 1 2 3 4 5 6 7 8 9 10 11 12

6, 7, 10, 6, 2, 8, 7, 7, 8 Find the first, second and third quartiles 2, 6, 6, 7, 7, 7, 8, 8, 10 Second = 7, 8, 8, 10 2, 6, 6, 7 First = Third = 1 2 3 4 5 6 7 8 9 10 11 12

Practice Answer the questions. Draw a Box-and-Whisker plot for each question.

Homework: page 371 problems 12 and 14 page 378 problems 16 and 18 Make a Stem-and-Leaf plot 12) 24, 29, 17, 50, 39, 51, 19, 22, 40, 45, 20, 18, 23, 30 14) 15, 39, 13, 31, 46, 9, 38, 17, 32, 10, 12, 45, 30, 1, 32, 23, 32, 41 Make a Box-and-Whisker plot 16) 10, 5, 9, 50, 10, 3, 4, 15, 20, 6 18) 8, 8, 10, 10, 1, 12, 8, 6, 5, 1, 9, 10

Warm Up Draw a Stem-and-Leaf plot Find the mean, median, mode and range Draw a Box-and-Whisker plot 1130, 695, 900, 220, 350, 500, 630, 180, 170, 145, 185, 140 Second Quartile = 285 Mean = 437.08 No mode First Quartile = 175 Median = 285 Range = 990 Third Quartile = 662.5

Homework: page 371 problems 12 and 14 page 378 problems 16 and 18 Make a Stem-and-Leaf plot 12) 24, 29, 17, 50, 39, 51, 19, 22, 40, 45, 20, 18, 23, 30 14) 15, 39, 13, 31, 46, 9, 38, 17, 32, 10, 12, 45, 30, 1, 32, 23, 32, 41 Make a Box-and-Whisker plot 16) 10, 5, 9, 50, 10, 3, 4, 15, 20, 6 18) 8, 8, 10, 10, 1, 12, 8, 6, 5, 1, 9, 10

Chapter 6 Quiz Review

Find the mean, median, mode and range 4, 8, 10, 6, 12, 16, 10, 22 Mean = 11 Median = 10 Mode = 10 Range = 18 What can you tell from the data?

Draw a Stem-and-Leaf Plot 60, 74, 75, 63, 78, 70, 50, 74, 52, 74, 65, 78, 54 Draw a Box-and-Whisker Plot 12, 5, 3, 8, 10, 7, 6, 5 Second Quartile = 6.5 First Quartile = 5 Third Quartile = 9

Quiz on Chapter 6

Bell Curve, Standard Deviation, Z-curve, etc. Statistics Bell Curve, Standard Deviation, Z-curve, etc.

Descriptive Statistics What’s Normal? Descriptive Statistics

What is standard deviation? - measures the spread of the data from the mean - What is the 68-95-99.7 rule in a normal distribution? Use your copy to shade in the regions shown.

The 68-95-99.7 Rule For a normal distribution, 68% of the data generally falls within 1 standard deviation of the mean. 95% of the data generally falls within 2 standard deviations of the mean. 99.7% of the data generally falls within 3 standard deviations of the mean.

Average of a set of data values Notation Sigma notation: sum of all the elements Average of a set of data values = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 Read as x bar sample = x1 + x2 + x3 + x4 + x5 Read as mu population

Mean Absolute Deviation Average of the DISTANCES between each data value and the mean

Variance Average of the squares of the differences between each data value and the mean

Square root of the variance Standard Deviation Square root of the variance

Measures of Dispersion describes the average distance from the mean describes the spread of the data

Investigating Dispersions Based on the Mean The SAT scores for ten students are given. The school wants to determine spread about the mean to fill out a report. 1026, 1150, 1153, 1157, 1161, 1206, 1253, 1258, 1285, 1311 Calculate the mean. = 1196

Investigating Dispersions Based on the Mean Create a chart of values for the SAT data set and determine the distance each data piece is from the mean. x 1026 1150 1153 1157 1161 1206 1253 1258 1285 1311 -170 -46 -43 -39 -35 10 57 62 89 115

Investigating Dispersions Based on the Mean What is the sum of the differences from the mean? -170 – 46 – 43 – 39 – 35 + 10 + 57 + 62 + 89 + 115 = Will this always happen? Test grades: 50, 70, 62, 80, 70, 76 What can be done to getting around the problem of always getting zero? Is there a way to get rid of the negatives?

Investigating Dispersions Based on the Mean Mathematically, we can take the absolute value of a number to ensure that it is positive. x 1026 1150 1153 1157 1161 1206 1253 1258 1285 1311 170 46 43 39 35 10 57 62 89 115

Investigating Dispersions Based on the Mean What is the sum of the absolute value distances? 666 The Mean Absolute Deviation = 66.6 80 The Standard Deviation =

The 68-95-99.7 Rule SAT problem 956 1036 1116 1196 1276 1356 1436 How many SAT scores fall within one standard deviation from the mean? 7 What % of the data does this represent? 70%

Homework How many texts do you send today?

Warm Up 2) 6m – 3 = 10 - 6(2 – m) 1) 26 – 9p = -1 -3 = -2 No solution 3) S = 2πrh, solve for h 4) Name the property (5 + x)6 = 6(5 + x) Commutative 5) Name the property 9 + 0 = 9 Identity

In a park that has several basketball courts a student samples the number of players playing basketball over a two week period and has the following data. 10 90 30 20 50 30 60 40 70 40 30 60 80 20

What is the mean for the data? 10 90 30 20 50 30 60 40 70 40 30 60 80 20

Distance from the mean 90 80 70 60 60 50 Mean = 45 40 40 30 30 30 20 10

90 What if we find the average of the difference between each data value and the mean? 80 45 70 35 60 60 25 15 15 50 5 Mean = 45 -5 -5 -15 -15 40 40 -15 -35 -25 -25 -35-15+5+15+25-15+35+45-25-15-5-5+15-25 = 30 30 30 20 20 10

90 What if we find the average of the DISTANCES from each data value to the mean? 80 45 70 35 60 60 15 25 15 50 35+15+5+15+25+15+35+45+25+15+5+5+15+25= 14 5 Mean = 45 280 14 5 5 15 =20 15 15 40 40 25 35 25 30 30 30 20 20 10

90 One Standard Deviation from the mean 80 70 =68.222 60 60 50 Mean = 45 40 40 =21.778 30 30 30 20 20 10

Calculating Mean Absolute Deviation How many texts did you send yesterday? 4 Calculate the mean absolute deviation for the data set. Calculate the standard deviation for the data set.

Calculating Standard Deviation How much time does it take for a dead cell phone battery to completely recharge?

Calculating Standard Deviation Mr. Bolling’s homework assignment for his students was to determine how much time it takes for their dead cell phone battery to completely recharge. The results for the amount of time (to the nearest quarter hour) for 20 students are shown below. 3.75 3.25 4 4.5 4.75 3.75 4 3.5 4.25 5 4.25 4 3.75 4.5 2.5 4 4.5 3.5 4 4.25

Calculating Standard Deviation What are the mean, mode, and median of the data? mean: 4, mode: 4, median: 4

Calculating Standard Deviation Calculate 1-Var Stats = 0.548 What does the standard deviation represent in this data?

Sample Question for A.9 Student Andy Bill Carrie Dan Ed Frank Gus Height 46 51 50 42 56 48 57 Henry Izzi Jack Ken Louise Manny Ned Owen 45 52 49 41 53 46 43 56 What is the approximate mean absolute deviation? D) 5 A) 3.4 B) 4.3 C) 4.5 What is the interpretation of the mean absolute value deviation of 4.3?

Sample Question for A.9 Student Andy Bill Carrie Dan Ed Frank Gus Height 46 51 50 42 56 48 57 Henry Izzi Jack Ken Louise Manny Ned Owen 45 52 49 41 53 46 43 56 Use your calculator to find the mean and standard deviation of the data set to the nearest inch? D) 50, 4.5 A) 49, 5 B) 50, 5.5 C) 49.5, 5.5

Warm Up 1) Find mean, median, mode and range: 56, 58, 63, 71, 78, 84, 85, 86, 82, 78, 65, 58 Mean = 72 Median = 74.5 Mode = 58 & 78 Range = 30 3) Rewrite so y is a function of x x = y + 3 2) y = x – 3

Descriptive Statistics Z-Scores for Algebra I Descriptive Statistics

How Close is Close? Activity

Z-score Position of a data value relative to the mean. Tells you how many standard deviations above or below the mean a particular data point is. z-score = describes the location of a data value within a distribution referred to as a standardized value Population Sample

In order to calculate a z-score you must know: a data value the mean the standard deviation

Z-scores Here are 23 test scores from Ms. Bienvenue’s stat class. 81 80 77 73 83 74 93 78 80 75 67 73 77 83 86 90 79 85 83 89 84 82 What is the mean score? 80.5 What is the standard deviation? 5.9

Z-scores Here are 23 test scores from Ms. Bienvenue’s stat class. 81 80 77 73 83 74 93 78 80 75 67 73 77 83 86 90 79 85 83 89 84 82 The bold score is Michele’s. How did she perform relative to her classmates? Michele’s score is “above average”, but how much above average is it?

Z-scores If we convert Michele’s score to a standardized value, then we can determine how many standard deviations her score is away from the mean. What we need: Michele’s score mean of test scores standard deviation 86 80.5 = 0.93 5.9 Therefore, Michele’s standardized test score is 0.93. Nearly one standard deviation above the class mean.

Michele’s Score = 86 Michele’s z-score = .93 62.2 68.3 74.4 80.5 86.6 92.7 98.8 -3 -2 -1 1 2 3

Consider this problem: Calculate a z-score Consider this problem: The mean salary for math teachers in Big State is $45,000 per year with a standard deviation of $5,000. The mean salary of a Piggly-Wiggly bagger is $21,000 with a standard deviation of $2,000.

Calculate a z-score Who has the better salary relative to the mean? A Big State teacher making 63,000 or a grocery bagger making 30,000? Teacher : 63,000 or Grocery Bagger: 30,000 What is the interpretation of the two z-scores? Who has a better salary relative to the mean?

4) Rewrite so y is a function of x Warm Up 1) 2) 5x – 2 = 8 5x = 10 x = 2 4) Rewrite so y is a function of x 3(y – x) = 10 – 4x 3) Name the property If a = b and b =c, then a = c 3y – 3x = 10 – 4x 3y = 10 – x Transitive Property

Sample Question for A.9 Student Andy Bill Carrie Dan Ed Frank Gus Height 46 51 50 42 56 48 57 Which students’ heights have a z-score greater than 1? A) All of them Mean = 50 Standard Deviation = 5.3 B) Bill, Carrie, Ed and Gus C) Ed and Gus D) None of them

Sample Question for A.9 Which students have a z-score less than -2? Andy Bill Carrie Dan Ed Frank Gus Height 46 51 50 42 56 48 57 Which students have a z-score less than -2? Mean = 50 Standard Deviation = 5.3 A) All of them B) Dan and Andy C) Only Dan D) None of them

Sample Question for A.9 Which student’s height has a z-score of zero? Andy Bill Carrie Dan Ed Frank Gus Height 46 51 50 42 56 48 57 Which student’s height has a z-score of zero? A) Bill B) Carrie C) Frank D) None of them

Sample Question for A.9 Given a data set with a mean of 125 and a standard deviation of 20, describe the z-score of a data value of 120? Mean = 125 Standard Deviation = 20 A) Less than -5 B) Between -5 and -1 C) Between -1 and 0 D) Greater than 0

Sample Question for A.9 Given a data set with a mean of 30 and a standard deviation of 2.5, find the data value associated with a z-score of 2? Mean = 30 Standard Deviation = 2.5 A) 36 B) 35 C) 34.5 D) 32.5

Sample Question for A.9 Suppose the test scores on the last exam in Algebra I are normally distributed. The z-scores for some of the students in the course were: 1.5, 0, -1.2, -2, 1.95, 0.5 1) List the z-scores of students that were above the mean. 1.5, 1.95, and 0.5

Sample Question for A.9 Suppose the test scores on the last exam in Algebra I are normally distributed. The z-scores for some of the students in the course were: 1.5, 0, -1.2, -2, 1.95, 0.5 2) If the mean of the exam is 80, did any of the students selected have an exam score of 80? Explain. One student with a z-score of 0.

Sample Question for A.9 Suppose the test scores on the last exam in Algebra I are normally distributed. The z-scores for some of the students in the course were: 1.5, 0, -1.2, -2, 1.95, 0.5 3) If the standard deviation of the exam was 5 and the mean is 80, what was the actual test score for the student having a z-score of 1.95? 90

Quiz