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Mean: Average Median: Middle of an ordered list Exact middle for an odd # of items Average of the middle two for an even # of items Mode: Most frequent Range: Highest - Lowest

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Helps you to see where the majority of the data lies, as each part is 25% of the data Lowest and highest values = endpoints Median of the data = center of the box Median of the lower part and upper part = edges of the box

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low Q1 median Q3 high lowest 25% 2 nd 25% 3 rd 25% highest 25% the box contains 50% of the data Outliers are 1.5. IQR from the ends of the box IQR = Q3 – Q2 Extreme Outliers are 3IQR from the ends of the box The high and the low are not always Outliers, not all data sets contain outliers.

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Relatively evenly distributed (normal) data Skewed left (longer left tail) Skewed right (longer right tail) Skew is determined by the tail

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Draw boxplot for the following test scores: 98, 75, 80, 74, 92, 88, 83, 60, 72, 99 Ordered list: 60, 72, 74, 75, 80, 83, 88,92, 98, 99 Draw a number line Plot the end points Find the median Find the median of the first half Find the median of the second half Draw the box around the three medians Connect the box with whiskers to the endpoints 60 70 80 90 100

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Displays all data Stem Leaf 1 st #(s) Last #

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Similar to a stem and leaf plot but does not necessarily retain the precise values of the data Given: 10, 18, 21, 26, 30, 31, 38, 40 Stem and LeafDot Plot 1 0, 8 2 1, 6 3 0, 1, 81 2 3 4 4 0

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10 2 5 7 20 1 6 30 5 8 9 9 40 2 3 5 7 8 50 2 60 3 6 the median the middle of the 17 values or 309 the first quartile the middle of the first half or (201+206)/2=203.5 the third quartile the middle of the second half or (407+408)/2=407.5 the inter-quartile range the difference of the quarter points 407.5-203.5=204 the mode the most frequent 309 the percentile for 305 305 if the 5 th item, 5/17=.294 * 100= 29.4 or the 29 th percentile the value closest to the 60 th percentile 60/100=x/17.6 = x/17.6*17 = x 10.2 = x the 10 th item (402) is closest to the 60 th Percentile Find the standard deviation enter all the data in L1 press STAT calc, choose one-var stat St. dev. = Ϭ x EXAMPLE: Given a stem and leaf plot FIND:

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Shows how many and approximate values of the data If the points follow a pattern, you can find the regression line

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Press 2 nd + 7 1 2 (clears everything) Press 2 nd 0 x -1 find diagnostics on press enter Press Stat enter Xs go in L1 Ys go in L2 Press Y=, arrow up press enter, zoom 9

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Decide what pattern the point appear to be following Press STAT arrow over to calc Choose the correct pattern 4 for linear 5 for quadratic 0 for exponential Press variable, arrow to y-vars, press 1, press 1, enter Write down the value of r Press Y= write down the equation Press graph to see the fit

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Predicting knowing x Set the window to be large enough for the given value Graph Press 2 nd trace (calc) Choose 1 (value) Enter the value and press enter Estimating knowing y Set the window to be large enough for the given value Enter the value in Y2= Press 2 nd trace (calc) Choose 5 (intersect) Press enter three times You may also substitute values into the equation

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Find the equation for the following data and determine the value when x = 2 and when x = 7 xy -5 0-2 10 31 43 54 66 Scatterplotenter data in stat edit Linear regression values Graph to make sure the line fits the pattern Use the calculations and enter a value of 2 Use the calculations and enter a value of 7 Click on the calculator to see how to find a regression line Now try it for your self, checking along the way to see if you have the same values/screen shots as belowclick each time you are ready to check your calculations.

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How can we determine all the possible outcomes of a given situation? TREE DIAGRAMan illustrative method of counting all possible outcomes. List all the choices for the 1 st event Then branch off and list all the choices for the second event for each 1 st event, etc.

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A restaurant offers a salad for $3.75. You have a choice of lettuce or spinach. You may choose one topping, mushrooms, beans or cheese. You may select either ranch or Italian dressing. How many days could you eat at the restaurant before you repeat the salad? Lettuce spinach mushrooms beans cheese mushrooms beans cheese ranch Italian ranch Italian ranch Italian ranch Italian ranch Italian ranch Italian

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While the tree diagram is beneficial in that it lists every possible outcome, the more options you have the more difficult it is to draw the diagram. Fundamental counting Principleis a mathematical version of the tree diagram, it gives the # of possible ways something can be accomplished but not a list of each way.

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Example: Jani can choose from gray or blue jeans, a navy, white, green or stripped shirt and running shoes, boots or loafers? How many outfits can she wear? _______ _______ _______ pants shirts shoes 233=18

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Permutationsall the possible ways a group of objects can be arranged or ordered Example: There are four different books to be placed in order on a shelf. A history book (H), a math book (M), a science book (S), and an English book (E). How many ways can they be arranged? 24 WAYS 4 3 2 1 = 24 H, M, S, E H, M, E, S H, S, E, M H, S, M, E H, E, M, S H, E, S, M M, E, S, H M, E, H, S M, S, H, E M, S, E, H M, H, E, S M, H, S, E S, M, E, H S, M, H, E S, H, M, E S, H, E, M S, E, M, H S, E, H, M E, M, S, H E, M, H, S E, H, M, S E, H, S, M E, S, M, H E, S, H, M

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A permutation of n objects r at a time follows the formula Example: This can be done on your calculator with the following keystrokes: Type the number before the P Press math Over to prb Choose number 2 nPr Enter the number after the P Press enter.

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How can you determine the difference between a permutation and a combination?

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Combinationsthe number of groups that can be selected from a set of objects --the order in which the items in the group are selected does not matter

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Example: How many three person committees can be formed from a group of 4 peopleJoe, Jim, Jane, and Jill Joe, Jim, Jill Joe, Jill, Jane Joe, Jim Jane Is Joe, Jane, Jim A different committee Jim, Jane, Jill Formula:

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What is the difference between replacement and repetition?

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This can be done on your calculator with the following keystrokes: Type the number before the C Press math Over to prb Choose number 3 nCr Enter the number after the C Press enter.

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Replacementusing the same object again (n r ) Example: The keypad on a safe has the digits 1- 6 on it how many: a) four digit codes can be formed _____ _____ b) four digit codes can be formed if no 2 digits can be the same _____ _____ 6666 6543

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Repetitionoccurs when you have identical items in a group Example: Find all arrangements for the letters in the word TOOL ____ ____ TOOLOLOTLOTO TOLOOLTOLOOT TLOOOTOLLTOO OTLO OOTL OOLT We would expect 24 but since you cant distinguish between the two Os all possibilities with the Os switched are removed 4321

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Formula for repetitions: where s and t represent the number of times an item is repeated EXAMPLE: How many ways can you arrange the letters in BANANAS A N The factorial key is also found my pressing math and arrowing over to PRB

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?2 1 3 4 Circular Permutationarranging items in a circle when no reference is made to a fixed point Example: How many ways can you arrange the numbers 1-4 on a spinner? We would expect 4! Or 24 ways but we only have 6 Circular permutations are always (n-1)! A1 2 3 4 B1 2 4 3 C1 3 2 4 D1 3 4 2 E1 4 2 3 B1 4 3 2 ?2 1 3 4 D

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If all outcomes are successful, the probability will be 1 If no outcomes are successful, the probability will be 0 So Probability is 0 P 1

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Examples: What is the probability of getting an ace from a deck of 52 cards? 4 aces so What is the probability of rolling a 3 on a 6 sided die? there is 1 3 on 6 sides so

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What is the probability of rolling an even number? 2,4, 6 are even so What is the probability of getting 2 spades when 2 cards are dealt at the same time? at the same time indicates use of a combination hint there are 13 spades

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What is the probability of getting a total of 5 when a pair of dice is rolled? +123456 1234567 2345678 3456789 45678910 56789 11 6789101112 Draw the following chart for the sum of all rolls and count how many have a sum of 5

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What is meant by compound probability?

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OR: P(A or B) = P(A) + P(B) – P(A and B) Example: What is the probability of getting a 2 or a 5 on the roll of a die? Exclusive Events: events that do not have bearing on each other

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What is the probability of drawing an ace or a heart? ace + heart – ace of hearts + - = Events are inclusive if they have overlap!

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AND: indicates multiplication Examples: What is the probability of tossing a three of the roll of a die and getting a head when you toss a coin? three and a head * = These events are independenthave no effect on the outcome of the other

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CS1512 Foundations of Computing Science 2 Lecture 20 Probability and statistics (2) www.csd.abdn.ac.uk/~jhunter/teaching/CS1512/lectures/ © J R W Hunter,

CS1512 Foundations of Computing Science 2 Lecture 20 Probability and statistics (2) www.csd.abdn.ac.uk/~jhunter/teaching/CS1512/lectures/ © J R W Hunter,

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