Math Review Data Analysis, Statistics, and Probability

Data Interpretation Your primary task in these questions is to interpret information in graphs, tables, or charts, and then compare quantities, recognize trends and changes in the date, or perform calculations based on the information you have found. Your primary task in these questions is to interpret information in graphs, tables, or charts, and then compare quantities, recognize trends and changes in the date, or perform calculations based on the information you have found. You should be able to understand info presented in a table or various types of graphs. You should be able to understand info presented in a table or various types of graphs.

Data Interpretation Circle Graphs (Pie Charts) Circle Graphs (Pie Charts) This pie chart shows the percentage of its total expenditures that Weston spends on various types of expenses. Suppose you are given that the total expenses in 2004 were $10 million, and you are asked for the amount of money spent on the police dept and fire dept combined. You can see that together these two categories account for 23% + 12%, so 35% total. Therefore, 35% of $10 million is $3.5 million. This pie chart shows the percentage of its total expenditures that Weston spends on various types of expenses. Suppose you are given that the total expenses in 2004 were $10 million, and you are asked for the amount of money spent on the police dept and fire dept combined. You can see that together these two categories account for 23% + 12%, so 35% total. Therefore, 35% of $10 million is $3.5 million.

Data Interpretation Line Graphs Line Graphs This graph shows the high and low temps in Weston for the first 7 days of Feb. This graph shows the high and low temps in Weston for the first 7 days of Feb. From the graph you can see that the high on Feb 5 was 25 degrees and the low was 10 degrees. You can also see the difference between high and low temps that day was 15. From the graph you can see that the high on Feb 5 was 25 degrees and the low was 10 degrees. You can also see the difference between high and low temps that day was 15.

Data Interpretation Bar graphs Bar graphs This one shows the amount of snow that fell each day in Weston for the first 7 days of Feb. For example, you can see that no snow fell on Feb 2 and that 6 inches of snow fell the next day. This one shows the amount of snow that fell each day in Weston for the first 7 days of Feb. For example, you can see that no snow fell on Feb 2 and that 6 inches of snow fell the next day.

Data Interpretation Pictographs Pictographs Presents data using pictorial symbols. Presents data using pictorial symbols. So you can see that 40 snowmen were built on Feb 7 and 20 were built the day before. So you can see that 40 snowmen were built on Feb 7 and 20 were built the day before.Date Number of Snowmen Feb 1 Feb 2 Feb 3 Feb 4 Feb 5 Feb 6 Feb 7 = 10 snowmen = 10 snowmen

Data Interpretation Scatterplot Scatterplot Compares two characteristics of the same group of people or things. You can see a lot of info from this plot. There are 20 people. 4 with 1 year of experience, 3 with 2 years, 4 with 3 years, 2 with 4 years, 3 with 5 years, and 4 with 6 years. The median level of experience is 3 years. Salary tends to increase with experience. 3 people make $700 a week and 3 others make $850 week. The median salary is $ Compares two characteristics of the same group of people or things. You can see a lot of info from this plot. There are 20 people. 4 with 1 year of experience, 3 with 2 years, 4 with 3 years, 2 with 4 years, 3 with 5 years, and 4 with 6 years. The median level of experience is 3 years. Salary tends to increase with experience. 3 people make $700 a week and 3 others make $850 week. The median salary is $787.50

Data Interpretation Questions may ask to read the info on the chart/graph, or to identify specific pieces of information (data), compare data from different parts of the graph, and manipulate the data. Questions may ask to read the info on the chart/graph, or to identify specific pieces of information (data), compare data from different parts of the graph, and manipulate the data. Make sure to Make sure to Look at the graph to make sure you understand it and what info is being displayed Look at the graph to make sure you understand it and what info is being displayed Read the labels Read the labels Make sure you know the units Make sure you know the units Understand what is happening to the data as you move through the table, graph, or chart. Understand what is happening to the data as you move through the table, graph, or chart. Read the question carefully. Read the question carefully.

Data Interpretation Example Example The graph below shows profits over time. The greater the profits the higher the point on the vertical axis will be. Each tick mark is another $1000. As you move right along the horiz. axis, months are passing. The graph below shows profits over time. The greater the profits the higher the point on the vertical axis will be. Each tick mark is another $1000. As you move right along the horiz. axis, months are passing.

Data Interpretation Example Example In what month or months did each company make the greatest profit? In what month or months did each company make the greatest profit? For company x it was April For company x it was April For company y it was May For company y it was May

Data Interpretation Example Example Between which two consecutive months did each company show the greatest increase in profit? Between which two consecutive months did each company show the greatest increase in profit? Increase/Decrease in profit is shown by the slope (steepness of line) Increase/Decrease in profit is shown by the slope (steepness of line) Just looking at it, you can tell for company x it was between march and april Just looking at it, you can tell for company x it was between march and april For company y it is between jan and feb For company y it is between jan and feb

Data Interpretation Example Example In what month did the profits of the 2 companies show the greatest difference? In what month did the profits of the 2 companies show the greatest difference? The point where they are farthest apart – so it would be in April The point where they are farthest apart – so it would be in April

Data Interpretation Example Example If the rate of increase or decrease for each company continues for the next 6 months at the same rate shown between April and May, which company would have higher profits at the end of that time? If the rate of increase or decrease for each company continues for the next 6 months at the same rate shown between April and May, which company would have higher profits at the end of that time? Need to extend the line between Apr and May Need to extend the line between Apr and May Can easily see the answer would be company y Can easily see the answer would be company y

Statistics Arithmetic Mean Arithmetic Mean Average Average Formula: Formula: (Sum of list of values)/(number of values in list) (Sum of list of values)/(number of values in list) Example: Example: Three kids, aged 6, 7, and 11. Find the Arithmetic mean. Three kids, aged 6, 7, and 11. Find the Arithmetic mean. (6+7+11)/(3) = 24/3 = 8 years (6+7+11)/(3) = 24/3 = 8 years

Statistics Median Median The middle value of a list when the numbers are in order. The middle value of a list when the numbers are in order. Place the values in order (ascending or descending) and select the middle value. Place the values in order (ascending or descending) and select the middle value. Example: Example: Find the median: 200, 2, 667, 19, 4, 309, 44, 6, 1 Find the median: 200, 2, 667, 19, 4, 309, 44, 6, 1 Order: 1, 2, 4, 6, 19, 44, 200, 309, 667 Order: 1, 2, 4, 6, 19, 44, 200, 309, 667 Middle value is the fifth, so median is 19 Middle value is the fifth, so median is 19 For an even list of numbers, average the two middle values to get the median For an even list of numbers, average the two middle values to get the median

Statistics Mode Mode The value or values that appear the greatest number of times. The value or values that appear the greatest number of times. Example: Example: Find the mode: 1, 5, 5, 7, 89, 4, 100, 276, 89, 4, 89, 1, 8 Find the mode: 1, 5, 5, 7, 89, 4, 100, 276, 89, 4, 89, 1, 8 89 appears 3 times (more than any other number) – it’s the mode! 89 appears 3 times (more than any other number) – it’s the mode! It is possible to have more than one mode. It is possible to have more than one mode.

Statistics Weighted Average Weighted Average The average of 2 or more groups that do not all have the same number of members. The average of 2 or more groups that do not all have the same number of members. Example: Example: 15 members of a class had an average (mean) SAT math score of 500. The remaining 10 members had an average of 550. What is the average score of the entire class? 15 members of a class had an average (mean) SAT math score of 500. The remaining 10 members had an average of 550. What is the average score of the entire class? Can’t take average of two numbers b/c of different amount of students in each group. Has to be weighted toward the group with the greater number. Can’t take average of two numbers b/c of different amount of students in each group. Has to be weighted toward the group with the greater number. So, multiply each average by its weighted factor first and then average them. So, multiply each average by its weighted factor first and then average them. ( (500*15)+(550*10) /25) = 520 ( (500*15)+(550*10) /25) = 520

Statistics Average of Algebraic Expressions Average of Algebraic Expressions Average same way as other values. Average same way as other values. Example: Example: What is the average (mean) of 3x+1 and x-3? What is the average (mean) of 3x+1 and x-3? Find the sum of the expressions and divide by the number of expressions: Find the sum of the expressions and divide by the number of expressions: ( (3x+1)+(x-3) )/2 = (4x-2)/2 = 2x-1 ( (3x+1)+(x-3) )/2 = (4x-2)/2 = 2x-1

Statistics Using Averages to Find Missing Numbers Using Averages to Find Missing Numbers You can use simple algebra in basic average formulas to find missing values when the average is given: You can use simple algebra in basic average formulas to find missing values when the average is given: Basic Average formula: Basic Average formula: (sum of a list of values)/(number of values in the list)=average (sum of a list of values)/(number of values in the list)=average If you have the average and the number of values, you can figure out the sum of the values: If you have the average and the number of values, you can figure out the sum of the values: Average*number of values=sum of values Average*number of values=sum of values

Statistics Example Example The average (mean) of a list of 10 numbers is 15. if one of the numbers is removed, the average of the remaining numbers is 14. what is the number that was removed? The average (mean) of a list of 10 numbers is 15. if one of the numbers is removed, the average of the remaining numbers is 14. what is the number that was removed? You can figure out the sum of all the values: 15*10=150 You can figure out the sum of all the values: 15*10=150 Sum of values for one removed: 14*9=126 Sum of values for one removed: 14*9=126 The difference between the 2 sums will give you the missing number =24. The difference between the 2 sums will give you the missing number =24.

Probability Some questions will involve elementary probability. Some questions will involve elementary probability. Example: Example: Find the probability of choosing an even number at random from the set: {6, 13, 5, 7, 2, 9} Find the probability of choosing an even number at random from the set: {6, 13, 5, 7, 2, 9} There are 6 numbers total and only 2 even numbers, so the probability would be 2/6 = 1/3 There are 6 numbers total and only 2 even numbers, so the probability would be 2/6 = 1/3 Remember, the probability of an event is a number between 0 and 1, inclusive. If an event is certain, it has probability of 1. If an event is impossible (cannot occur) the probability is 0. Remember, the probability of an event is a number between 0 and 1, inclusive. If an event is certain, it has probability of 1. If an event is impossible (cannot occur) the probability is 0.

Probability Independent/Dependent Events Independent/Dependent Events Two events are independent if the outcome of either event has no effect on the other. (Toss a penny and it has ½ landing on heads and then toss a nickel and it has ½ prob of landing on heads). Two events are independent if the outcome of either event has no effect on the other. (Toss a penny and it has ½ landing on heads and then toss a nickel and it has ½ prob of landing on heads). To find the probability of 2 or more ind. events occurring together, you multiply the different probabilities together. (1/2 * ½)= ¼ To find the probability of 2 or more ind. events occurring together, you multiply the different probabilities together. (1/2 * ½)= ¼ If the outcome of one event affects the probability of another event, they are dependent events. You must use logical reasoning to help figure out probabilities involving dependent events. If the outcome of one event affects the probability of another event, they are dependent events. You must use logical reasoning to help figure out probabilities involving dependent events. So the penny landing on heads is ½ but of it then landing on tails is 0 (can’t land on tails at the same time) So the penny landing on heads is ½ but of it then landing on tails is 0 (can’t land on tails at the same time)

Probability Example Example On Monday, Anderson High School’s basketball team will play the team from Baker High School. On Wednesday, Baker’s team will play the team from Cole High School. On Friday, Cole will play Anderson. In each game, either team has a 50 percent chance of winning. A) What is the probability that Anderson will win both its games? B) What is the probability that Baker will lose both its games? C) What is the probability that Anderson will win both its games and Baker will lose both its games? On Monday, Anderson High School’s basketball team will play the team from Baker High School. On Wednesday, Baker’s team will play the team from Cole High School. On Friday, Cole will play Anderson. In each game, either team has a 50 percent chance of winning. A) What is the probability that Anderson will win both its games? B) What is the probability that Baker will lose both its games? C) What is the probability that Anderson will win both its games and Baker will lose both its games? A) Anderson has ½ prob of winning 1 st game and also 2 nd game. Ind events. So just multiply: ½ * ½ = ¼ A) Anderson has ½ prob of winning 1 st game and also 2 nd game. Ind events. So just multiply: ½ * ½ = ¼ B) Baker has ½ prob of losing 1 st game and also 2 nd game. Ind events. So just multiply: ½ * ½ = ¼ B) Baker has ½ prob of losing 1 st game and also 2 nd game. Ind events. So just multiply: ½ * ½ = ¼ C) For Anderson to win both games and Baker to lose both, three games must be played out: And beats Baker, Cole beats Baker, And beats Cole. Each game has ½ prob, and ind. So just multiply all together: ½ * ½ * ½ = 1/8 C) For Anderson to win both games and Baker to lose both, three games must be played out: And beats Baker, Cole beats Baker, And beats Cole. Each game has ½ prob, and ind. So just multiply all together: ½ * ½ * ½ = 1/8

Probability Geometric Probability Geometric Probability Some probability questions in the math section may involve geometric figures. Some probability questions in the math section may involve geometric figures. Example: Example: The large circle has a radius 8 and small circle has radius 2. If a point is chosen at random from the large circle, what is the probability that the point chosen will be in the small circle? The large circle has a radius 8 and small circle has radius 2. If a point is chosen at random from the large circle, what is the probability that the point chosen will be in the small circle? Area of large circle is 64pie, and area of small circle is 4pie Area of large circle is 64pie, and area of small circle is 4pie Area of small circle/ area of large circle = 4pie/64pie = 4/64 = 1/16 Area of small circle/ area of large circle = 4pie/64pie = 4/64 = 1/16 So probability of choosing a point from the small circle is 1/16 So probability of choosing a point from the small circle is 1/16