Chapter 2 Statistics. Mean, Median, Mode, and Range Definitions

Transcription

1 M a t h C h a p t e r 2 S t a t i s t i c s P a g e 1 of 16 Chapter 2 Statistics Mean, Median, Mode, and Range Definitions Mean : The "Mean" is computed by adding all of the numbers in the data together and dividing by the number elements contained in the data set. Example : Data Set = 2, 5, 9, 3, 5, 4, 7 Number of Elements in Data Set = 7 Mean = ( ) / 7 = 5 Median : The "Median" of a data set is dependant on whether the number of elements in the data set is odd or even. First reorder the data set from the smallest to the largest then if the number of elements are odd, then the Median is the element in the middle of the data set. If the number of elements are even, then the Median is the average of the two middle terms. Examples : Odd Number of Elements Data Set = 2, 5, 9, 3, 5, 4, 7 Reordered = 2, 3, 4, 5, 5, 7, 9 Median = 5 Examples : Even Number of Elements Data Set = 2, 5, 9, 3, 5, 4 Reordered = 2, 3, 4, 5, 5, 9 Mode : The "Mode" for a data set is the element that occurs the most often. It is not uncommon for a data set to have more than one mode. This happens when two or more elements occur with equal frequency in the data set. A data set with two modes is called bimodal. A data set with three modes is called trimodal. Examples : Single Mode Data Set = 2, 5, 9, 3, 5, 4, 7 Mode = 5 Examples : Bimodal Data Set = 2, 5, 2, 3, 5, 4, 7 Modes = 2 and 5 Range : The "Range" for a data set is the difference between the largest value and smallest value contained in the data set. First reorder the data set from smallest to largest then subtract the first element from the last element. Examples : Data Set = 2, 5, 9, 3, 5, 4, 7 Reordered = 2, 3, 4, 5, 5, 7, 9 Range = ( 9-2 ) = 7 Median = ( ) / 2 = 4.5

2 M a t h C h a p t e r 2 S t a t i s t i c s P a g e 2 of 16 Reading a stem and leaf plot A veterinarian weighed all the dogs that visited his office last month. Dog s weights (lbs) Stem Leaf Rewrite the values below: Sometimes one stem and leaf plot can contain two sets of data Lengths of salmon (mm) Male Female How long was the shortest male? How long was the longest female? What is the mean of female salmon? What is the mean of this distribution? What is the mode of this distribution? What is the median of this distribution? Practice! 1. The train conductor made sure to count the number of passengers on each train. Passengers per train Stem Leaf How many trains had at least 23 passengers but fewer than 92 passengers?

3 M a t h C h a p t e r 2 S t a t i s t i c s P a g e 3 of 16 Mean Deviation How do you find the mean deviation? Step 1: Find the mean of the distribution Step 2: Find the deviation from the mean - Subtract the mean of the distribution from the value - Make sure your answer is in absolute value (no negative numbers) Step 3: Add up all the deviations from the mean and divide by how many values you have (find the mean of the deviations). Example #1: Find the mean deviation of the following data set: 2, 4, 7, 9, 10, 15, 16 Step 1: Step 2: Step 3: Value Deviation from the mean Absolute Value The mean of the distribution is. The Mean Deviation is.

4 M a t h C h a p t e r 2 S t a t i s t i c s P a g e 4 of 16 Example #2: Find the mean deviation of the following distribution: 1, 25, 26, 27, 28, 29, 60 Step 1: Step 2: Step 3: Value Deviation from the mean Absolute Value The mean of the distribution is. The Mean Deviation is. Example #3: Find the mean deviation of the following distribution: 20, 20, 20, 20, 20, 20

5 M a t h C h a p t e r 2 S t a t i s t i c s P a g e 5 of 16 Percentile Percentiles divide a distribution into one hundred groups that each contains 1% of the data values. Percentile 1 1 % Of the data Percentile 2 1 % Of the data Percentile 3 1 % Of the data Percentile 98 1 % Of the data Percentile 99 1 % Of the data Percentile % Of the data Round up! If your answer has a decimal in it you must round the number up to a whole number. Some examples; To find the percentile of a number: Percentile of x = Round up! Example A: What is the percentile rank of 28 in the distribution below? Step #1: How many values are less than x (in this case 28)? Nless = Step #2: How many values are equal to x (in this case 28)? Nequal = Step #3: How many values are there in total, including x (in this case 28)? Ntotal = Step #4: Plug your values into the formula and solve. Step #5: Round up (always!) when finding percentile rank. Percentile rank =

6 M a t h C h a p t e r 2 S t a t i s t i c s P a g e 6 of 16 When you know the percentile: Round down! Example B: Given the list of data values below, which one has a percentile rank of at least 72? 5, 7, 7, 9, 12, 15, 15, 18, 19, 24, 27, 29, 30, 35 Step 1: Find the total number of data values in the distribution Ntotal = Step 2: Insert percentile and Ntotal into formula and solve Step 3: Round down Step 4: There are data values less than the date value with a percentile rank of at least x (in this case 72). So count the less. 5, 7, 7, 9, 12, 15, 15, 18, 19, 24, 27, 29, 30, 35 ** You don t need to know all of the values to solve a percentile question. has a percentile rank of at least 72. Example C: The results of 198 students who wrote a math test are listed below in increasing order. 48, 48, 49, 79, 79, 80, 81, 81, results 35 results Kim s result was 80. What was her percentile rank? Step 1: How many results are less than x (in this case 80)? Step 2: How many results are equal to x (in this case 80)? Step 3: How many results are there in total? Nless = Nequal = Ntotal = Step 4: Plug your values into the formula and solve. Step #5: Round up (always!) when finding percentile rank. Answer:

7 M a t h C h a p t e r 2 S t a t i s t i c s P a g e 7 of 16 Correlation When the two sets of data are strongly linked together we say they have a High Correlation. Correlation is Positive when the values increase together, and Correlation is Negative when one value decreases as the other increases Correlation can have a value: 1 is a perfect positive correlation 0 is no correlation (the values don't seem linked at all) -1 is a perfect negative correlation The value shows how good the correlation is (not how steep the line is), and if it is positive or negative. Example: Ice Cream Sales vs Temperature Temperature C Ice Cream Sales 14.2 $ $ $ $ $ $ $ $ $ $ $ $408 The local ice cream shop keeps track of how much ice cream they sell versus the temperature on that day, here are their figures for the last 12 days. The table on the left represents the data and below is a scatter plot which also represents the data.

8 M a t h C h a p t e r 2 S t a t i s t i c s P a g e 8 of 16 Correlation When we are placing different correlation coefficients in order from weakest to strongest, we can ignore whether they are positive or negative because +1 is equal to -1 when thinking of the strength of correlation. A) For example, place the following in order from weakest to strongest correlation: 0.2, 0.95, -0.5, B) Place the following correlation coefficients in order from strongest to weakest: -1, 0.77, 0.25, -0.6, 0.85 Just by looking at a scatter plot, we can guess what the correlation coefficient will be. We can also use measurements to determine whether our guess was accurate. Determining the correlation coefficient. Step 1: Draw a line through the middle of the data, trying to follow the slope of the data. This line should represent most of the points. Step 2: Construct the smallest rectangle around all the significant points on the scatter plot with two sides parallel to the line. Step 3: Find the correlation coefficient using the following formula: If the correlation is positive If the correlation is negative ) )

9 M a t h C h a p t e r 2 S t a t i s t i c s P a g e 9 of 16 Correlation: Regression Line The regression line is the line that best represents all of the points within a correlation. Mayer-Line Method Used to predict an unknown value from a table of values or a scatterplot Ex: Determine the equation of the regression line from the following distribution: x y P 1 (x 1,y 1 ) P 2 (x 2,y 2 ) Steps for Mayer-Line Method - Divide data into two groups - Find the mean of the coordinates - Use these two points to build y ax b - Use rule to solve for an unknown value If we were to add another point to this situation, what would y be if x was 20?

10 M a t h C h a p t e r 2 S t a t i s t i c s P a g e 10 of 16 Example: The table below shows the relationship between the age of a tree and the circumference of its trunk. Tree Growth Age Circumference (cm) a) Determine the equation of the regression line associated with this situation. b) What should be the circumference of a 25-year-old tree trunk be? c) If a tree trunk s circumference is cm, what should its age be?

11 M a t h C h a p t e r 2 S t a t i s t i c s P a g e 11 of 16 Statistics Practice Questions 1. A doctor carried out a statistical study to see if there is a relationship between the height of 42 boys on their 20 th birthday and their respective height on their 3 rd birthday. The following table shows the data gathered. Height of a boy on his 3 rd birthday (cm) Height of a boy on his 20 th birthday (cm) [80, 84[ [84, 88[ [88, 92[ [92, 96[ [96, 100[ [155, 162[ [162, 169[ [169, 176[ [176, 183[ [183, 190[ Which of the following statements best describes the linear correlation between the variables of this study? A) The linear correlation is negative and weak. B) The linear correlation is negative and strong. C) The linear correlation is positive and weak. D) The linear correlation is positive and strong. 2. Given the four scatter plots below. Dependent variable Scatter Plot 1 Dependent variable Scatter Plot 2 Independent variable Independent variable Dependent variable Scatter Plot 3 Dependent variable Scatter Plot 4 Independent variable Independent variable In which of the following arrangements are the scatter plots ordered from highest to lowest correlation? A) 3, 2, 4, 1 C) 2, 3, 4, 1 B) 2, 4, 3, 1 D) 1, 4, 3, 2

12 M a t h C h a p t e r 2 S t a t i s t i c s P a g e 12 of Which of the following correlation coefficients represents the strongest correlation? A) -0.8 C) 0.5 B) -0.3 D) The coefficient of linear correlation between two variables is -1. Which of the following statements is true? A) There is no correlation. B) There is a very low correlation, with one variable increasing as the other increases. C) There is a very low correlation, with one variable decreasing as the other increases. D) There is a perfect correlation, with one variable decreasing as the other increases. 5. The data listed below represent the marks obtained by 32 students on a mathematics test What is the percentile rank of a student with a mark of 66%? The percentile rank is. 6. Given the list of data values below, what is the percentile rank of 61? 49, 52, 55, 61, 72, 74, 75, 76, 76, 77, 81, 85, 87, 91

13 M a t h C h a p t e r 2 S t a t i s t i c s P a g e 13 of Determine the mean deviation for the following table. Stem Leaf 4 1, 3, , 7, 8, 8 7 2, 5 8. Miriam applied for admission into the police technology program. Her results on an oral exam, a written exam and a physical condition exam will determine her acceptance. There were 25 candidates that took these exams. Here are Miriam's results on each exam. Oral Exam Physical Condition Exam 33 Placed 7 th out of Here are the conditions for admission: The average on the oral exam must be greater than 80%. The percentile rank on the physical condition exam must be 70 or higher. Will Miriam be accepted into the police technology program?

14 M a t h C h a p t e r 2 S t a t i s t i c s P a g e 14 of The following scatter plot represents the relationship between two variables. y x What is the value of the correlation coefficient? 10. A school held a track and field competition. The table below lists the height jumped by 14 competitors in the high jump event, according to the students' age. Age (years) Height Jumped According to Age Height (cm) Age (years) Height (cm) Given the above results, how high could a 19-year-old child be expected to jump?

15 M a t h C h a p t e r 2 S t a t i s t i c s P a g e 15 of The number of errors made on the same test was recorded for two groups of students. Indicate which group had the lowest mean deviation. Group A: 0, 0, 0, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 7, 8, 8, 9, 10, 10, 11 Group B: Number of Frequency Number of Frequency errors errors A hospital required its nurses to write a Nursing exam on technology. Of the 265 nurses who wrote this exam: Karena is the only one who scored 139 points. 176 nurses scored fewer than 139 points. 88 nurses scored more than 139 points. In what percentile is Karena s score?

16 M a t h C h a p t e r 2 S t a t i s t i c s P a g e 16 of 16 Chapter 2 Statistics Statistics Practice Questions ANSWERS 1. D 2. C (Remember negative and positive don t have anything to do with the strength of the correlation) 3. A 4. D 5. The percentile rank is The percentile rank is The mean deviation is Example of an appropriate method of solution Average on the oral exam % = 82.5% 40 Thus her average is above 80% Miriam meets the first condition. Percentile rank Miriam placed 7 th out of 25. This means that 18 students had a lower score than she did. Miriam's percentile rank is: Miriam is in the 74 th percentile and meets the second condition. Answer Yes, Miriam will be admitted into the program. 9. The correlation coefficient is -0.8 (values ranging from to are accepted) 10. A 19 year old could be expected to jump about cm. 11. Group B had the lower mean deviation (Group B had a mean deviation of 2.06 while Group A had a mean deviation of 2.35) 12. Karena is in the 67 th percentile.