Algebra 2. Outliers. Measures of Central Tendency (Mean, Median, Mode) Standard Deviation Normal Distribution (Bell Curves)

Transcription

1 Algebra 2 Outliers Measures of Central Tendency (Mean, Median, Mode) Standard Deviation Normal Distribution (Bell Curves)

2 Algebra 2 Notes #1 Chp 12 Outliers In a set of numbers, sometimes there will be "outliers". WHAT IS AN OUTLIER? Outliers are stragglers (extremely high or extremely low values) in a data set that can throw off your stats. For example, if you were measuring children s nose length, your average value might be thrown off if Pinocchio was in the class. Example 1: Given {1, 99, 100, 101, 103, 109, 110, 201} In this set of random numbers, and are outliers. 1 is an extremely low value and 201 is an extremely high value. Example 2: Given the numbers {25, 29, 3, 32, 85, 33, 27, 28}, find any outliers Answer: In this set, there are 2 outliers: and Example 3: Answer: The outlier is: Example 4: Answer: The outlier is: Example 5: Complete the line plot Given the data set {19, 24, 19, 21, 21, 16, 6, 22, 19, 28}, draw a line plot and find any outliers Answer: Example 6: Given the data set {7, 12, 8, 11, 13, 15, 14, 16, 7, 9}, draw a line plot and find any outliers. Complete the line plot Answer: Note: There will not always be an outlier.

3 Measures of Central Tendency Statistics is the branch of mathematics that deals with the collection, organization, analysis, and interpretation of numerical data. The following are 3 commonly used statistics, commonly known as measures of central tendency: 1. The MEAN, or average, of n numbers is the sum of the numbers divided by n. The mean is denoted by, which is read as "x bar." x For the data x 1, x 2, x 3,..., x n the mean is x = x 1 +x 2 +x x n n a) What is the mean of the numbers 8, 9, 13, and 18? 2. The MEDIAN of n numbers is the middle number when the numbers are written in order (from least to greatest). If there are 2 middle numbers, you will have to find the mean. a) What is the median of the numbers 5, 8, 9, 13, and 18? b) What is the median of the numbers 5, 8, 9, 10, 13, and 18? 3. The MODE of n numbers is the number of numbers that occur most frequently. There may be one mode, no mode, or more than one mode. What is the mode of the numbers 5, 8, 9, 13, and 9? What is the mode of the numbers 15, 8, 6, 15, and 6? What is the mode of the numbers 2, 8, 7, 13, and 9? Measures of Central Tendency tell you what the center of the data is. Other commonly used statistics are called Measures of Dispersion. They tell you how spread out the data numbers are. One simple measure of dispersion is the RANGE, which is the difference between the greatest data value and the least data value. (Range = highest # lowest#) In statistics, dispersion is the extent to which a distribution of numbers is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range.

4 How to complete "Finding Mean" Worksheet. The first 2 problems from the worksheet are shown below. Complete these same 2 problems on your worksheet and continue showing the same work for the rest of the problems. Also, complete the puzzle at the bottom of the worksheet. Remember to round answers to the nearest tenth.

5 Algebra 2 Chp 12 Notes #2 Normal Distribution/Bell Curve/Standard Deviation Data can be "distributed" (spread out) in different ways. It can spread out more on the right, more on the left, or it can be all jumbled up. But there are many cases where the data tends to be around a central value with no bias left or right, and it gets close to a "Normal Distribution" like this: "Bell Curve" Normal Distribution We say the data is "normally distributed." This distribution curve is often called a "Bell Curve" because it looks like a bell. Many things closely follow a Normal Distribution: > heights of people > blood pressure > marks on a test The Normal Distribution (Bell Curve) has the following attributes: > (if you can draw a line down the center of the picture and > symmetry about the center. get 2 similar halves, it's symmetrical.) > mean = median = mode Half of the data values (50%) are less than the mean (the average of all the numbers) > Half of the data values (50%) are greater than the mean (the average of all the numbers) The bell curve is seen in tests like the ACT and SAT. The bulk of students will score the average (C). Smaller numbers of students will score a B or D, and an even smaller percentage of students will score an A or an E. This creates a distribution that resembles a bell (hence the nickname), or bell curve. Mean Median Mode

6 Normal Distributions/Bell Curves/Standard Deviation x = mean (average of the data set) 2(.0015)+2(.0235)+2(.135)+2(.34) = 1 =100% (of the data given) This bell curve or "normal distribution"curve tells you what percentage of your data falls within a certain number of standard deviations from the mean (the average of the set): 68% of the data falls within one standard deviation of the mean. 95% of the data falls within two standard deviations of the mean. 99.7% of the data falls within three standard deviations of the mean. Note: Areas under this curve can also be represented as percentages. Areas under this curve can also represent probabilities. The Standard Deviation is a measure of how spread out numbers are within a set of data values. The Greek letter sigma,, denotes standard deviation. σ Normal Distribution Curve (Bell Curve) σ 68% +σ ( =.68 = 68%) 68% of data values are within 1 standard deviation ( ) of the mean +σ 2σ 95% +2σ ( =.95 = 95%) 95% of data values are within 2 standard deviations ( ) of the mean +2σ ( =.997 = 99.7%) 3σ 99.7% +3σ 99.7% of data values are within 3 standard deviations ( ) of the mean +3σ

7 c b a 1. What percent of the area under a normal curve lies within a. 1 standard deviation of the mean? b. within 2 standard deviations of the mean? c. within 3 standard deviations of the mean? Give the percent of the area under a normal curve represented by the shaded region

8 Estimate the standard deviation of each graph: x How to Compute the mean( x) and standard deviation( σ): Enter the data values (use the following buttons) 2nd STAT 1 VAR 2 VAR ENTER DATA x 1 = Type in each number, then press the down arrow key to get to x 2 and repeat as necessary. To find the answers, press STAT VAR You will see the following: n: this is how many numbers there are in the set x: this is the mean (average) Sx: (ignore this one) σx: this is the standard deviation To stay in STAT Mode and enter a new data set: 2nd STAT 1 VAR 2 VAR CLRDATA ENTER DATA x 1 = To exit STAT Mode: or Compute the mean( ) and standard deviation( σ) of each. Round answers to the nearest tenth: x 5. {5, 7, 9, 11, 15} 6. {20, 21, 20, 24, 27} x = σ (mean) (standard deviation) = x = σ = 1

9 7. {2, 5, 3, 6, 10} 8. {15, 12, 17, 22, 25, 27, 20} 9. {4, 7, 16, 24, 33} 10. {30, 35, 31, 38} 11. {26, 27, 20, 22, 20} 12. {42, 36, 42, 37, 51} 1

10 Test scores are normally distributed with a mean of 70% (C average) and a standard deviation of 10%. Approximately how many students scored an A or B in a class of: 13) 25 students 14) 38 students 15) 20 students 16) 50 students Approximately how many students scored a B in a class of: 17) 20 students 18) 50 students Approximately how many students scored an A in a class of: 19) 20 students 20) 50 students Approximately how many students scored a D in a class of: 21) 20 students 22) 50 students Approximately how many students scored an E in a class of: 23) 20 students 24) 50 students 1

11 A normal distribution has a mean of 62 and a standard deviation of 4. Find the probability (keep in decimal form) that a randomly selected x value is in the given interval. 25) between 58 and 66 26) between 62 and 74 27) between 50 and 70 28) 62 or greater A normal distribution has a mean of 10 and a standard deviation of 1. Find the probability (keep in decimal form) that a randomly selected x value is in the given interval. 29) between 8 and 12 30) between 7 and 13 31) between 8 and 11 32) 12 or greater 1

12 Algebra 2 Outliers Measures of Central Tendency (Mean, Median, Mode) Standard Deviation Normal Distribution (Bell Curves)

13 Algebra 2 Notes #1 Chp 12 Outliers In a set of numbers, sometimes there will be "outliers". WHAT IS AN OUTLIER? Outliers are stragglers (extremely high or extremely low values) in a data set that can throw off your stats. For example, if you were measuring children s nose length, your average value might be thrown off if Pinocchio was in the class. Example 1: Given {1, 99, 100, 101, 103, 109, 110, 201} In this set of random numbers, and are outliers. 1 is an extremely low value and 201 is an extremely high value. Example 2: Given the numbers {25, 29, 3, 32, 85, 33, 27, 28}, find any outliers Answer: In this set, there are 2 outliers: and Example 3: Answer: The outlier is: Example 4: Answer: The outlier is: Example 5: Complete the line plot Given the data set {19, 24, 19, 21, 21, 16, 6, 22, 19, 28}, draw a line plot and find any outliers Answer: Example 6: Given the data set {7, 12, 8, 11, 13, 15, 14, 16, 7, 9}, draw a line plot and find any outliers. Complete the line plot Answer: Note: There will not always be an outlier.

14 Measures of Central Tendency Statistics is the branch of mathematics that deals with the collection, organization, analysis, and interpretation of numerical data. The following are 3 commonly used statistics, commonly known as measures of central tendency: 1. The MEAN, or average, of n numbers is the sum of the numbers divided by n. The mean is denoted by, which is read as "x bar." x For the data x 1, x 2, x 3,..., x n the mean is x = x 1 +x 2 +x x n n a) What is the mean of the numbers 8, 9, 13, and 18? 2. The MEDIAN of n numbers is the middle number when the numbers are written in order (from least to greatest). If there are 2 middle numbers, you will have to find the mean. a) What is the median of the numbers 5, 8, 9, 13, and 18? b) What is the median of the numbers 5, 8, 9, 10, 13, and 18? median = 9 median = 3. The MODE of n numbers is the number of numbers that occur most frequently. There may be one mode, no mode, or more than one mode. What is the mode of the numbers 5, 8, 9, 13, and 9? What is the mode of the numbers 15, 8, 6, 15, and 6? What is the mode of the numbers 2, 8, 7, 13, and 9? Measures of Central Tendency tell you what the center of the data is. Other commonly used statistics are called Measures of Dispersion. They tell you how spread out the data numbers are. One simple measure of dispersion is the RANGE, which is the difference between the greatest data value and the least data value. (Range = highest # lowest#) In statistics, dispersion is the extent to which a distribution of numbers is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range.

15 How to complete "Finding Mean" Worksheet. The first 2 problems from the worksheet are shown below. Complete these same 2 problems on your worksheet and continue showing the same work for the rest of the problems. Also, complete the puzzle at the bottom of the worksheet. Remember to round answers to the nearest tenth

16 Algebra 2 Chp 12 Notes #2 Normal Distribution/Bell Curve/Standard Deviation Data can be "distributed" (spread out) in different ways. It can spread out more on the right, more on the left, or it can be all jumbled up. But there are many cases where the data tends to be around a central value with no bias left or right, and it gets close to a "Normal Distribution" like this: "Bell Curve" Normal Distribution We say the data is "normally distributed." This distribution curve is often called a "Bell Curve" because it looks like a bell. Many things closely follow a Normal Distribution: > heights of people > blood pressure > marks on a test The Normal Distribution (Bell Curve) has the following attributes: > (if you can draw a line down the center of the picture and > symmetry about the center. get 2 similar halves, it's symmetrical.) > mean = median = mode Half of the data values (50%) are less than the mean (the average of all the numbers) > Half of the data values (50%) are greater than the mean (the average of all the numbers) The bell curve is seen in tests like the ACT and SAT. The bulk of students will score the average (C). Smaller numbers of students will score a B or D, and an even smaller percentage of students will score an A or an E. This creates a distribution that resembles a bell (hence the nickname), or bell curve. Mean Median Mode

17 Normal Distributions/Bell Curves/Standard Deviation 2(.0015)+2(.0235)+2(.135)+2(.34) = 1 =100% (of the data given) x = mean (average of the data set) This bell curve or "normal distribution"curve tells you what percentage of your data falls within a certain number of standard deviations from the mean (the average of the set): 68% of the data falls within one standard deviation of the mean. 95% of the data falls within two standard deviations of the mean. 99.7% of the data falls within three standard deviations of the mean. Note: Areas under this curve can also be represented as percentages. Areas under this curve can also represent probabilities. The Standard Deviation is a measure of how spread out numbers are within a set of data values. The Greek letter sigma,, denotes standard deviation. σ Normal Distribution Curve (Bell Curve) σ 68% +σ ( =.68 = 68%) 68% of data values are within 1 standard deviation ( ) of the mean +σ 2σ 95% +2σ ( =.95 = 95%) 95% of data values are within 2 standard deviations ( ) of the mean +2σ ( =.997 = 99.7%) 3σ 99.7% +3σ 99.7% of data values are within 3 standard deviations ( ) of the mean +3σ

18 1. What percent of the area under a normal curve lies within a. 1 standard deviation of the mean? b. within 2 standard deviations of the mean? c. within 3 standard deviations of the mean? =.68= 68% 95% 99.7% Give the percent of the area under a normal curve represented by the shaded region. 50% % =0.5=50% 47.5% =.1585=15.85% 27% =.475=47.5% 2.5% =.27=27% 49.85% =.025=2.5% =.4985=49.85% 15.85% 68% =.1585=15.85% =.68=68% 95% 61% =.95=95% =.61=61%

19 Estimate the standard deviation of each graph: How to Compute the mean( x) and standard deviation( σ): Enter the data values (use the following buttons) 2nd STAT 1 VAR 2 VAR ENTER DATA Type in each number, then press the down arrow key to get to x 2 and repeat as necessary. To find the answers, press STAT VAR You will see the following: n: this is how many numbers there are in the set x: this is the mean (average) Sx: (ignore this one) σx: this is the standard deviation To stay in STAT Mode and enter a new data set: 2nd STAT 1 VAR 2 VAR CLRDATA ENTER DATA x 1 = To exit STAT Mode: Compute the mean( ) and standard deviation( ) of each. Round answers to the nearest tenth: x 5. {5, 7, 9, 11, 15} 6. {20, 21, 20, 24, 27} x = σ (mean) or (standard deviation) = 47 5 x 1 = x = σ = σ

20 7. {2, 5, 3, 6, 10} 8. {15, 12, 17, 22, 25, 27, 20} {4, 7, 16, 24, 33} 10. {30, 35, 31, 38} 11. {26, 27, 20, 22, 20} 12. {42, 36, 42, 37, 51}

21 Test scores are normally distributed with a mean of 70% (C average) and a standard deviation of 10%. Approximately how many students scored an A or B in a class of: 13) 25 students 14) 38 students 15) 20 students 16) 50 students Approximately how many students scored a B in a class of: 17) 20 students 18) 50 students Approximately how many students scored an A in a class of: 19) 20 students 20) 50 students Approximately how many students scored a D in a class of: 21) 20 students 22) 50 students Approximately how many students scored an E in a class of: 23) 20 students 24) 50 students 1

22 A normal distribution has a mean of 62 and a standard deviation of 4. Find the probability (keep in decimal form) that a randomly selected x value is in the given interval. 25) between 58 and =.68 26) between 62 and = ) between 50 and 70 28) 62 or greater =.5 A normal distribution has a mean of 10 and a standard deviation of 1. Find the probability (keep in decimal form) that a randomly selected x value is in the given interval. 29) between 8 and =.95 30) between 7 and = ) between 8 and 11 32) 12 or greater = =.025 1