Calculate the mode, median and range for a set of data Decide which average best represents the set of data

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1 5. The mode, median and range Calculate the mode, median and range for a set of data Decide which average best represents the set of data Key words average data mode median range An average gives information that is typical of a set of data. The mode is a type of average. In a set of data it is the value or values that happen most often: 6 9 The mode is in this set of numbers as is the value that occurs most often. Data that is not about numbers (e.g. colours of car, exam grades) can only have the mode as an average. The median is another type of average. In an ordered set of data it is the middle value: 6 9 The median is in this set of numbers as it is the middle value. If there is an even number of values in the set of data, the median is halfway between the middle two values. The range is not an average. It is the smallest value subtracted from the largest and shows how spread out the data is: 6 9 The range is 8 in this set of numbers, because 9 8 Example The ages in years of a group of people are: What is a) the mode b) the median c) the range of their ages? d) Which average would you use to represent the data? Explain why. e) Another 0 year old joins the group. Calculate the new mode, median and range. Put the ages in order: a) The mode is 5 b) The median is c) The range is 8. d) I would use the median, as it is halfway between the lower and upper ages. The mode in this case is the highest value so does not represent the younger people in the group. e) There are now modes: 0 and 5 years. The median is years. The range is still 8. There are more 5 s than any other number. is the middle number. The range is the largest smallest value (5 7 8) The ages in order are now The median is half way between 0 and. 6 Maths Connect R

2 Exercise Ravi asks his friends what mark they got for their last piece of homework a) What is the median mark? b) What is the mode? c) What is the range? d) Ravi wants to include his mark of 7. What is the new median mark? e) Which of the averages would you use to represent the data? Explain why. The table shows the number of days that rain fell each month in London: Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Number of days Find the mode, median and range of the data over the year. Which of these cannot have a median value, only a mode? a) Age b) How long a person can hold his or her breath c) Amount in s spent on trainers d) Colour of school uniform e) Weight f) How long it takes to run 00 m g) Type of pet owned h) Favourite snack. Information that is not to do with numbers (called categorical data) can only have the mode as an average. The table below shows the typical temperature each month for Summertown. Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec C a) What is the range of the temperatures? b) What is the median temperature? Which consecutive whole numbers have a median of 8 and a range of 6?,, 5 are consecutive and have a median of and a range of. Which consecutive whole numbers have a total of and a median of? Five dice are thrown. The numbers showing have a mode and median of and a range of. Find all the possible solutions. Lynn, Mac, Toby and Mair have a combined age of 70 years. The modal age is years and the median age is 8 years. a) What are the ages of the people? b) Which average would you use to represent the data? Why? Which consecutive whole numbers have a total of 60 and a range of? Make a table using the headings below. Data which can only have a mode e.g. Hair colour Data which can have a mode and median value e.g. Height Write a sentence describing the difference in the type of data for the two headings. The mode, Number median : Proportion and range 7

3 5. The mean Calculate the mean for a set of data Key words mean range average sum The mean is an average. Like the median, it can only be used for data that is numerical. In everyday language when people say average they are usually talking about the mean. the sum of all the data The mean is calculated as: the number of items of data Example The best 8 long jump distances of an athlete were (in metres) Find the mean distance. The mean the total distance jumped the number of jumps m Exercise The number of people attending Premiership football matches at the beginning of 00 were: 8 096, 88,, 0 6, 0 0, 67 60, 5 7, 890 Find the mean attendance per match. The temperature at dawn for ten days in March were: 5 0 Find the mean temperature. Find the missing numbers shown by. They are all whole numbers, greater than 0. Each set of numbers has a mean of 6. a) 7 7 b) 5 5 c) or 8 Maths Connect R

4 Kate asks her friends how many times they were late for school last term. She puts her results in a spreadsheet. A B C Number of times Ordered number of late for school times late for school Total 8 Mean a) Look at column B. What will Kate enter in cell B7 to find the total number of times her friends were late? b) What will Kate enter in cell B8 to find the mean number of times her friends were late? c) Explain how she could use column C to find the median, mode and range of her data. a) If the mean of four numbers is 0, what is the sum of the four numbers? b) If the mean of ten numbers is 7. what is the sum of the ten numbers? Natacha needs a mean average of 85% in her four tests for an A* grade. So far her marks are 87% and 79%. What is the minimum average mark needed in the final two tests to get an A*? The mean sum of the numbers amount of numbers Start by finding the total sum she needs in the tests for a mean of 85%. A set of numbers has a mean of 5 and a total sum of 50. How many numbers are in the set? The table below shows the countries with the greatest number of cars. Which of the four countries has the most number of cars per head of population? Country Number of cars (millions) Population (millions) USA 7 77 Japan 7 Germany 0 86 Italy 0 58 Investigation Make a list showing how many days there are in each month during a year that is not a leap year. Find the mode and the median number of days in a month. What is the mean number of days in a month in a leap year? What is the mean number of days in a year over a period of four years? The mean 9

5 5. Frequency tables Record data in frequency tables Use frequency tables to calculate the mean Key words frequency event frequency table mean The frequency of an event is the number of times that event occurs. A frequency table is a way of organising this data. To find the mean, instead of adding up all the bits of data separately, we can find the totals for each group. Example Thirty people were asked how many pets they owned. These are the results: 0 5 a) Draw a frequency table to show this data. b) Use the table to work out the mean number of pets owned. a) Number of Tally Frequency Total number Record the results in pets owned of pets owned a tally column people own pets, so 8 pets in total The number of pets owned in total. total number of pets owned b) Mean 6 0 total frequency 0 The number of people asked. (total frequency) Exercise The frequency table below shows the number of letters for the first 50 words of a book Complete the table to find the mean length of word. Number of letters Frequency Number of letters frequency First find the total frequency and the total number of letters frequency. 50 Maths Connect R

6 A teacher records the number of pupils who are absent each day for a month. a) Put this information into a frequency table. b) Find the mean number of absent pupils each day Jake is collecting information about how many complete books his friends read in one week. He records the results for 5 friends. Unfortunately he has lost some of the numbers. Can you help him? Number of books Frequency Number of books frequency Total Total a) Copy and complete the frequency table. b) Calculate the mean number of books read in one week. The table below shows the number of children in each house in one particular street: Number of children per house 0 Frequency State whether each of the following statements is true or false: a) The modal number of children is b) The range of children is The modal number is a different c) The total number of children is 50 way of asking what the mode, the most common value, is. d) The number of houses in the street is 50 e) The mean number of children is. f) 7 houses contain children. A primary school has 8 classes consisting of the following number of pupils: 5, 6, 7, 6, 9, 9, 7, 9 Class size Frequency a) Draw a frequency table for this data with the headings: b) Find the mean number of pupils per class. c) The mean class size must be 0 or less. What is the greatest number of pupils that can join the school for this to be so? Sunita plays hockey for her school team. The tally chart shows the number of goals she has scored in matches this season. a) Copy and complete the frequency table. b) Find her mean number of goals per match. c) Her mean score after the next game changes to.6 goals per match. How many goals did Sunita score in the next game? The frequency column must add up to 5 as his survey was for a total of 5 friends. Number of goals scored Tally Frequency in a match 0 Frequency tables 5

7 5. Interpreting diagrams Interpret diagrams, graphs and charts Draw conclusions and find simple statistics from diagrams Key words bar chart compound bar chart bar-line graph pie chart Bar charts, compound bar charts, bar-line graphs and pie charts can all be used to represent data. Showing data in this way makes it easier to interpret and spot trends. Example a) What is the range of the number of i) boys ii) girls attending school? b) Is there enough information on the diagram to find the median value? c) The red dotted line shows the mean value. Show that the mean number of students attending school each day is 8. d) How does the diagram help to show that the mean is 8? e) Is there a modal value? Frequency Compound bar chart showing the number of pupils attending school Monday Tuesday Wednesday Day Thursday Friday Boys Girls a) i) The range is 5 ii) The range is 5. b) Yes. Putting the values in order gives , so the median is 8 pupils. ( ) c) d) There is a total of ( from Wednesday and from Thursday) above the mean line. Below the line, there is also a total of from the tops of the bars to the mean line ( from Tuesday, from Friday). As these are equal this shows that the mean is 8. e) No, as no value appears more than once. 5 Maths Connect R

8 Exercise 5.:... Meena draws a bar-line graph showing the number of goals scored by the school hockey team. a) Which of the three averages can be calculated from the graph? Calculate any averages that can be found. b) Calculate the range of goals scored. c) In week 7 the team scores 5 goals. Calculate the new averages and the range. 6 5 Bar-line graph showing the number of goals each week 0 week week week week week 5 week 6 The bar chart shows the grades for a class in their last mathematics test. a) Explain why only the mode can be found from the diagram. b) Can the range be calculated? c) How many pupils took the test? Frequency Bar chart showing grades for maths test A B C D Grade E a) Which city had the month Comparative bar chart showing the number with the most days of rain? of days of rain for cities in China b) Which month was this? 0 c) Which city had the month 8 with the fewest days of 6 rain? d) Which month was this? e) How many days of rain 0 did each city have in total J F M A M J J A S O N D during the year? Month f) Which city has the greatest range of days of rain? g) What is the mean number of days of rain for each of the two cities? Number of days Pupils in Year 7 were asked which sport was their favourite. a) Which sports were liked equally by both boys and girls? b) Which sport was preferred by girls? By boys? c) How many pupils were asked in total? d) How many girls were asked? e) How many boys were asked? Comparative bar chart showing favourite sport Swimming Cricket Table tennis Hockey Tennis Number of pupils Guangzhou Beijing Choose a bar graph or pie chart from a newspaper, magazine or travel brochure. Make up some questions about the information it shows for a partner to answer. Include questions about any averages that can be found. Girls Boys Interpreting diagrams 5

9 5.5 Describing probabilities Understand and use the probability scale from 0 to Know that the probability of an event not happening is ( probability of the event happening) Key words event outcome probability chance impossible certain An event (such as rolling a dice) can have different outcomes. When an ordinary six-sided dice is rolled, the possible outcomes are,,,, 5 or 6. Probability is a measure of chance. A probability can be expressed using words or numbers. When probability is expressed as a number it must be between 0 and. 0 impossible unlikely even likely certain Probability can be written as a fraction, a decimal or a percentage. the number of ways the outcome can happen Probability of an outcome the total number of possible outcomes Example In a game of scrabble there are eight letters remaining: This means that each e a t v t i s t letter has the same A letter is chosen at random. chance of being chosen. What is the probability it is a) t b) a vowel c) s or v d) b e) i f) not an i? a) 8 b) 8 c) 8 or d) 0 e) 8 An event either happens or it doesn t happen. The probability of an event happening the probability of it not happening equals, so 8 (probability of an i) 7 8 f) 8 7 Exercise 5.5:... Decide if each of the following is a fair way to start a game using a dice. If it is not fair, which player is most likely to start each time? a) b) c) d) Player Player Odd Even 6 Not a 6 Less than or more or 6,, or 5 Starting a game is fair if both players have the same chance of starting. 5 Maths Connect R

10 All the letters from the word BRILLIANCE are written on separate pieces of paper and put in a bag. One letter is taken out without looking. What is the probability that the letter will be a) a vowel b) L c) Z d) A, B or C e) I f) Not I? Show all your answers as a fraction, a decimal & a percentage. A bag of counters contains 0 counters: red and 6 blue. A counter is chosen at random. It is red. What is the probability that the next counter chosen is red if a) the counter is replaced into the bag b) the counter is not replaced? The probability of a train being late is 0.6. What is the probability that it will not be late? Mandy has accidentally put two old batteries back into a packet that also contains six new ones. She picks out a battery. a) What is the probability that it will work? b) What is the probability that it will not work? c) How many batteries should she take out to be certain that at least one will work? Copy and complete the spinner so that the probability of getting an odd number is and the probability of getting a is. A bag contains 0 cubes of four different colours. The probability of choosing each of the colours is shown. How many of each colour cube are there? Colour Red Green Probability 5 Yellow 9 0 White 0 A bag contains a number of counters. The probability of choosing a green counter is 0.. a) Explain why it is not possible for there to be eight counters. b) Harry takes two counters from the bag. They are both green. What is the smallest possible number of counters in the bag? A bag contains ten blue, nine white and six red cubes. a) A cube is taken out of the bag at random. Find the probability of each of the different colours being chosen as a decimal. b) A cube is removed from the bag. The probability of a blue cube is now 5, a white is and a red is. What colour was removed from the bag? Explain your answer. Investigation Draw a probability line showing the probability of six different events happening tomorrow. Tomorrow I will: Impossible Certain Go to the moon Describing probabilities 55

11 5.6 Probabilities of events Find all the possible outcomes of one event Find all the possible outcomes of two events happening together Record all possible outcomes using a table Key words random outcome event If a letter is chosen from the alphabet at random (every letter has the same chance of being chosen) there are many possible outcomes. Different outcomes may have different probabilities. For example, the probability of getting a vowel is less than that of getting a consonant. When two or more events happen together, such as tossing a coin and throwing a dice, it is useful to organise all of the outcomes in a table so that no outcomes are missed. Example If a letter is chosen at random from the word REVERSE there are four possible outcomes choosing R E, V and S. A letter is chosen at random. What is the probability it is a) an R? b) an E? c) a V? d) a vowel? e) a consonant? a) probability of an R 7 d) probability of a vowel 7 b) probability of an E 7 e) probability of a consonant 7 c) probability of a V 7 Example Helen spins a -sided spinner and throws a dice. a) List all the possible outcomes using a table. b) Use the table to find the following probabilities: i) a red on the spinner ii) a 6 on the dice iii) a on the dice and white on the spinner iv) not a blue v) white on the spinner and a 5 or 6 on the dice. a) Blue Red White, blue, red, white, blue, red, white, blue, red, white, blue, red, white 5 5, blue 5, red 5, white 6 6, blue 6, red 6, white 8 or ii) 8 or 6 iii) 8 b) i) 6 iv) 8 or v) Maths Connect R

12 Exercise Twenty cards are numbered from to 0. If a card is chosen at random: a) What is the probability of choosing an odd number? b) What is the probability of choosing a number less than 5? c) What is the probability of choosing a number divisible by? d) What is the probability of choosing a number in the range of to 5 inclusive? A school sells two colours of tickets for a raffle. Pink tickets are numbered from to 50. Blue tickets are numbered from to 00. All of the tickets are sold. If the tickets are drawn at random find the probabilities of drawing: a) a number from to 50 b) an odd number c) a blue ticket d) a number with two or three digits. Draw a table to show the outcomes when these two spinners are spun together. Spinner Spinner Draw a table to show the outcomes when a 5-sided spinner and a coin are spun together. Use the table to find the following probabilities: a) The spinner landing on blue. b) The coin landing on tails. Spinner c) The spinner not landing on white or red. d) The spinner landing on green and the coin landing on heads. e) The spinner landing on blue or black and the coin landing on tails. f) The spinner not landing on red and the coin not landing on heads. Copy and complete the tables below to show the total when two different pairs of dice are thrown and the numbers showing are added. Pair : ordinary dice Which pair of dice give the greatest probability of getting a total of: a) i) ii) 8 iii) 6 iv) an even amount v) less than 6? b) List the probabilities for your answers to part a). Investigation Pair : dice from 0 5 Two normal 6 dice are thrown and the numbers showing are multiplied together. Player A wins if the answer is even, otherwise player B wins. Play this game with a partner 0 times, recording who wins each time in a frequency table. Draw a table to investigate the different outcomes. Is the game fair? Coin Probabilities of events 57