Mathematics. Thomas Whitham Sixth Form S J Cooper

Transcription

1 Mathematics Handling Data Revision Notes For Year 8 Thomas Whitham Sixth Form S J Cooper. Probability of a single event. Probability of two events 3. Statistics Qualitative data 4. Statistics Time series data 5. Statistics Quantitative data 6. Statistical measures 7. Scatter diagrams 8. Conversion graphs

2 ~ ~. Probability Probability is a number between 0 and expressed either as a fraction or as a decimal. IMPOSSIBLE EVEN CHANCE CERTAIN LESS LIKELY MORE LIKELY Probability through symmetry If in an experiment an event A can occur in x ways and the total number of possible outcomes is N, then the probability that A occurs is given by: x P A N Toss a coin once [experiment]. Let A be the event that the coin falls heads. Here is the set S of all possible outcomes S = {H, T} i.e. there are possible outcomes. Here the set A = {H} only outcome P A Throw a die once [experiment]. Let A be the event that the number shown is prime. Here S = {,, 3, 4, 5, 6}...6 outcomes And A = {, 3, 5} P A 3 6

3 ~ 3 ~ Probability through statistical evidence In the previous 4 weeks the traffic lights at Barden lane were against me 4 mornings out of 0 weeks. What is the probability, based on this information that I will have to stop at the lights tomorrow morning? Stopped 4 times out of 0 times 4 7 P(stopped tomorrow) = 0 0 Alice bought 5 tickets in a raffle. If 00 tickets are sold, what is the probability that Alice will win first prize? P( st prize) = Probability of two events Probability tree diagrams A coin is tossed twice. Let A be the event that we get two heads. st toss nd toss outcome Head Head, Head Head Tail Head, Tail Tail Head Tail Tail, Head Tail, Tail

4 ~ 4 ~ P(Head, Head) = 4 Alan and Bob have a habit of being late to school with probabilities and 3 respectively. a) Complete the tree diagram below Alan Late Bob Late... a) b) What is the probability of one of the two boys being late to school tomorrow? Alan Not Late Late Not Late b) P(one late) = 6 Bob Late Not late Late Not late One late = 3 = 6 One late = 3 = 6

5 ~ 5 ~ Probability Space diagram A die is thrown twice. Let A be the event that the sum of the numbers shown is equal to 7. Here we use a two way table to generate possible outcomes Number on st die Number on nd die Each block represents a possible outcome and each shaded block represents a score of 7 on the two dice. Number of possible outcomes = 36 Number of sevens = 6 P A

6 Lunchtime arrangements ~ 6 ~ 3. Statistics Diagrammatic representation of Qualitative (descriptive) data Lunchtime arrangements of a class of 30 pupils are as follows: School Dinner 8 Sandwiches 9 Home 3 (i) Bar charts Don t forget to label axes, give a title, space equally the bars. Use capital printing and be neat at all times. Home Sandwiches School Dinner frequency

7 ~ 7 ~ The barchart below shows the amount of rainfall, in cm, there was in a small village during the first week in October. a) How much rainfall was there on Wednesday? b) Which day saw the least amount of rain? c) How much rainfall was there in the entire first week of October? a) Amount of rainfall on Wednesday = 6cm b) Least amount of rainfall was on Tuesday. c) Total amount of rainfall = = 86cm

8 ~ 8 ~ (ii) Pictograms Using the information on the lunchtime arrangements we could draw the following pictogram HOME = 5 PUPILS SANDWICHES SCHOOL DINNERS (iii) Pie chart Using the information on the lunchtime arrangements we could draw a pie chart by first finding the three angles that would fill the 360 of a circle. Because there are 30 pupils we must multiply each amount by as 30 x = 360 School Dinner 8 8x = 6 Sandwiches 9 9x = 08 Home 3 3x = 36

9 ~ 9 ~ Home (3) Sandwiches (9) School Dinner (8) The table below is the forms of transport taken by 36 workers of a factory Walk Cycle Bus Train Car Since the total is 36 person represents = 0

10 ~ 0 ~ Car () Train (9) Walk (3) Bus (4) Cycle (8) Include raw data and not the angle values in the sectors 4. Time series data The number of shirts sold in a shop each day of a certain week were as follows: Monday Tuesday Wednesday Thursday Friday Saturday

11 frequency ~ ~ Monday Tuesday Wednesday Thursday Friday Saturday Day of week 5. Diagrammatic representation of quantitative (numerical) data There are two types of quantitative data Discrete and Continuous data. Discrete data is data which you count. i.e. peas in a pod, goals scored in football. This data is usually ungrouped data. Continuous data is data which you measure. i.e. height of people in a class, amount of petrol in cars. This data is usually grouped data. Ungrouped data In a test marked out of 0, fifty primary school children obtained the following marks:, 3, 5, 4, 3, 5, 5, 4, 3, 6, 5, 4, 7, 4, 5, 3, 4, 4, 5, 5, 5, 4, 3, 4, 3 4, 3, 4, 5, 4, 3, 6, 3,, 6,, 6, 6, 3, 5,, 7, 5, 7,, 7, 6, 5, 8, 6

12 Frequency ~ ~ Firstly these results need to be placed into a tally chart then they can be represented in a vertical line graph Mark Tally Frequency l l l l l 3 3 l l l l l l l l 0 4 l l l l l l l l l 5 l l l l l l l l l l 6 l l l l l l 7 7 l l l l 4 8 l Mark

13 ~ 3 ~ 5 primary children sit a spelling test marked out of 50 and obtained the following results: 4, 34, 8,, 6, 7, 48, 9,, 36, 44, 50, 9,, 7, 30, 4, 47, 6, 8, 9,, 37, 48, 4. Represent these results in a stem and leaf diagram. Key 7 represents Once we have constructed the stem and leaf diagram we must then place leafs in numerical order

14 ~ 4 ~ Grouped data The playing time for 40 CDs are given in the table below. Playing time (minutes) Frequency We represent this data by histogram. frequency Playing time

15 ~ 5 ~ 6. Statistical measures Mean, Mode, Median There are three measures of average to be considered in the study of statistics, the (Arithmetic) Mean, the Median and the Mode. Ungrouped data The mean of ungrouped data is given by the formula: Sum of observations Mean number of observations This is the average of everyday use and is often simply referred to as the average. Jocky threw scores of 0, 40, 60, 5, 3, 0, 0, 5 in nine throws of one dart. Sum = = Mean 7 9 The median is the value of the middle observation when the observations have been written down in order of size. Taking Jock s scores and writing in order 3, 5, 5, 0, 0, 0, 40, 60, 60 Middle score Median = 0 When there is an even number of observations there will not be a middle one, but a middle two we take the median to be the mean of the middle two.

16 ~ 6 ~ For the six observations 5, 7,, 6, 9, 33 Median is between the and 6 Median = 4 The mode is the value which occurs most often in the observations. Taking Jock s scores and writing in order 3, 5, 5, 0, 0, 0, 40, 60, 60 The mode = 0 because this value occurs more often than any other value. Frequency distributions The mean of a frequency distribution is given by the formula: Sum of 'fx' column Mean Sum of 'f' column The median is as before The mode is that value (of x) with the highest frequency (f) The following number of goals were scored throughout a season by the Thomas Whitham Sixth Forms st team.,,,, 5, 0,,, 4, 3, 0,, 3, 0,,,, 4, 3, 3,,,,,

17 ~ 7 ~ Goals Scored Tally Frequency fx (x) (f) 0 l l l 3 0 l l l l l 6 6 l l l l l l l l l l l l 4 4 l l 8 5 l 5 Total 5 49 Sum of 'fx' column 49 Mean.96 Sum of 'f' column 5 Median = the 3 th value = Mode = value of x with highest frequency = Grouped data The first table shows the distribution of ages of pensioners who are members of a social club. The second table is constructed for calculations of the mean age. The values of x are mid-interval values.

18 ~ 8 ~ Age Frequency Age Frequency (f) (x) (f) fx Total Sum of 'fx' column 95 Mean years Sum of 'f' column 40 The mean will be an approximation since exact ages are not given, just age range. The mode is given as a modal class which is the one group with the highest frequency = years. Range is usually measured by the spread given by the difference between the greatest value and the least value. Jocky s scores 3, 5, 5, 0, 0, 0, 0, 40, 60, 60 has range given by: Range = 60 3 = 57

19 ~ 9 ~ 7. Scatter diagrams Brian s teacher asked him to measure the diameter D and Circumference C of 8 circular objects. He recorded his observations, plotted points on a scatter diagram and draw a line of best fit. D C y x Line of best fit Brian s teacher asked him to use his diagram to find the circumference for a diameter of 4cm. Brian drew the dotted line and was able to give an answer of cm (approximately)

20 ~ 0 ~ 8. Conversion Graphs The following straight line graph is to be used for conversion between C and F. It was constructed by plotting points from the following information. Freezing point Boiling point Boiling point of water of alcohol of water C F 3 67 The dotted line is the construction for converting 50 C into F F C

21 50 C = 0 F from the graph ~ ~

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