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Statistics N5 LS www.mathsrevision.com Mode, Mean, Median and Range Quartiles Semi-Interquartile Range ( SIQR ) Boxplots – Five Figure Summary Sample Standard Deviation Exam questions 15-May-17 Created by Mr. Lafferty 1 Starter Questions N5 LS www.mathsrevision.com 1. Find the value of x if 3x -3= x +5 42 o 2. Calculate x: 3. What is the chance of picking a three card from a pack of cards. x 1 4. 12 % of 480 2 15-May-17 Created by Mr Lafferty Maths Dept o Statistics Averages www.mathsrevision.com N5 LS Learning Intention Success Criteria 1. We are revising the terms mean, median, mode and range. 15-May-17 1. Understand the terms mean, range, median and mode. 2. To be able to calculate mean, range, mode and median. Created by Mr Lafferty Maths Dept Statistics Finding the mode www.mathsrevision.com N5 LS The mode or modal value in a set of data is the data value that appears the most often. For example, the number of goals scored by the local football team in the last ten games is: 2, 1, 2, 0, 0, 2, 3, 1, 2, 1. What is the modal score? 2. Is it possible to have more than one modal value? Is15-May-17 it possible to haveCreated no bymodal value? Yes Mr Lafferty Maths Dept Yes Statistics The mean N5 LS www.mathsrevision.com The mean is the most commonly used average. To calculate the mean of a set of values we add together the values and divide by the total number of values. Mean = Sum of values Number of values For example, the mean of 3, 6, 7, 9 and 9 is 367 99 5 15-May-17 34 5 Created by Mr Lafferty Maths Dept 6.8 Statistics Finding the median www.mathsrevision.com N5 LS The median is the middle value of a set of numbers arranged in order. For example, Find the median of 10, 7, 9, 12, 7, 8, 6, Write the values in order: 6, 7, 7, 8, The median is the middle value. 15-May-17 Created by Mr Lafferty Maths Dept 9, 10, 12. Statistics Finding the median www.mathsrevision.com N5 LS When there is an even number of values, there will be two values in the middle. For example, Find the median of 56, 42, 47, 51, 65 and 43. The values in order are: 42, 43, 47, 51, 56, There are two middle values, 47 and 51. 47 + 51 98 = = 49 2 2 15-May-17 Created by Mr Lafferty Maths Dept 65. Statistics Finding the range www.mathsrevision.com N5 LS The range of a set of data is a measure of how the data is spread across the distribution. To find the range we subtract the lowest value in the set from the highest value. Range = Highest value – Lowest value When the range is small; the values are similar in size. When the range is large; the values vary widely in size. 15-May-17 Created by Mr Lafferty Maths Dept Statistics The range N5 LS www.mathsrevision.com Here are the high jump scores for two girls in metres. Joanna Kirsty 1.62 1.59 1.41 1.45 1.35 1.41 1.20 1.30 1.15 1.30 Find the range for each girl’s results and use this to find out who is consistently better. Kirsty is consistently better ! 15-May-17 Joanna’s range = 1.62 – 1.15 = 0.47 Kirsty’s range = 1.59 – 1.30 = 0.29 Created by Mr Lafferty Maths Dept Frequency Tables Working Out the Mean www.mathsrevision.com N5 LS Example : This table shows the number of light bulbs used in people’s living rooms No of Bulbs (c) Freq. (f) 1 7 7x1=7 2 5 5 x 2 = 10 3 5 5 x 3 = 15 4 2 2x4=8 5 1 1x5=5 Totals 20 Adding a third column to this table will help us find the total number of bulbs and the ‘Mean’. Mean Number of bulbs 45 = 2.25 bulbs per room 20 15-May-17 Created by Mr. Lafferty Maths Dept. (f) x (B) 45 Statistics Averages www.mathsrevision.com N5 LS Now try N5 TJ Lifeskills Ex 24.1 Ch24 (page 232) 15-May-17 Created by Mr Lafferty Maths Dept Lesson Starter www.mathsrevision.com N5 LS Q1. Explain why 2.5% of £800 = £20 Q2. Calculate sin 90o Q3. Factorise 5y2 – 10y Q4. A circle is divided into 10 equal pieces. Find the arc length of one piece of the circle if the radius is 5cm. 15-May-17 Created by Mr. Lafferty 12 Quartiles www.mathsrevision.com N5 LS Learning Intention 1. We are learning about Quartiles. Success Criteria 1. Understand the term Quartile. 2. Be able to calculate the Quartiles for a set of data. 15-May-17 Created by Mr. Lafferty Maths Dept. Statistics Quartiles www.mathsrevision.com N5 LS Quartiles :Splits a dataset into 4 equal lengths. Median 25% 15-May-17 25% 50% 75% Q1 Q2 Q3 25% 25% Created by Mr Lafferty Maths Dept 25% Statistics Quartiles www.mathsrevision.com N5 LS Note : Dividing the number of values in the dataset by 4 and looking at the remainder helps to identify quartiles. R1 means to can simply pick out Q2 (Median) R2 means to can simply pick out Q1 and Q3 R3 means to can simply pick out Q1 , Q2 and Q3 R0 means you need calculate them all 15-May-17 Created by Mr Lafferty Maths Dept www.mathsrevision.com N5 LS Semi-interquartile Range (SIQR) = ( Q3 – Q1 ) ÷ 2 = ( 9– 3) ÷ 2 =3 Statistics Quartiles Example 1 : For a list of 9 numbers find the SIQR R1 3, 3, 7, 8, 10, 9, 1, 5, 9 9 ÷ 4 = 2 1 3 2 numbers 3 5 7 2 numbers 1 No. Q2 Q1 8 9 15-May-17 the 2nd and 3rd numbers 3 7 the 5th number the 7th and 8th number. 9 Created by Mr Lafferty Maths Dept 10 2 numbers 2 numbers The quartiles are Q1 : Q2 : Q3 : 9 Q3 Statistics Semi-interquartile Range Quartiles (SIQR) = ( Q3 – Q1 ) ÷ 2 = ( 10 – 3 ) ÷ 2 = 3.5 www.mathsrevision.com N5 LS Example 3 : For the ordered list find the SIQR. R3 3, 6, 2, 10, 12, 3, 4 7 ÷ 4 = 1 2 3 1 number 4 1 number Q1 15-May-17 3 6 10 1 number Q2 The quartiles are Q1 : the 2nd number 3 Q2 : the 4th number 4 Q3 : the 6th number. 10 Created by Mr Lafferty Maths Dept 12 1 number Q3 Statistics Averages www.mathsrevision.com N5 LS Now try N5 TJ Lifeskills Ex 24.2 Ch24 (page 236) 15-May-17 Created by Mr Lafferty Maths Dept Lesson Starter N5 LS www.mathsrevision.com In pairs you have 3 minutes to explain the steps of the Proportion Process 15-May-17 Created by Mr. Lafferty 19 Semi-Interquartile Range www.mathsrevision.com N5 LS Learning Intention 1. We are learning about Semi-Interquartile Range. Success Criteria 1. Understand the term SemiInterquartile Range. 2. Be able to calculate the SIQR. 15-May-17 Created by Mr. Lafferty Maths Dept. Inter-Quartile Range www.mathsrevision.com N5 LS The range is not a good measure of spread because one extreme, (very high or very low value can have a big effect). Another measure of spread is called the Semi - Interquartile Range and is generally a better measure of spread because it is not affected by extreme values. SIQR Q3 Q1 2 Finding the Semi-Interquartile range. Example 1: Find the median and quartiles for the data below. 6, 3, 9, 8, 4, 10, 8, 4, 15, 8, 10 Order the data Q2 Q1 3, 4, 4, 6, Lower Quartile = 4 8, 8, Median = 8 Q3 8, 9, 10, 10, Upper Quartile = 10 Inter- Quartile Range = (10 - 4)/2 = 3 15, Finding the Semi-Interquartile range. Example 2: Find the median and quartiles for the data below. 12, 6, 4, 9, 8, 4, 9, 8, 5, 9, 8, 10 10, 12 Order the data Q2 Q1 4, 4, 5, 6, Lower Quartile = 5½ 8, 8, Q3 8, Median = 8 9, 9, 9, Upper Quartile = 9 Inter- Quartile Range = (9 - 5½) = 1¾ Statistics www.mathsrevision.com N5 LS Now try N5 TJ Lifeskills Ex 24.3 Ch24 (page 237) 15-May-17 Created by Mr Lafferty Maths Dept Lesson Starter N5 LS www.mathsrevision.com In pairs explain term Gradient and how it can be linked to Pythagoras Theorem 15-May-17 Created by Mr. Lafferty 25 ( www.mathsrevision.com N5 LS Boxplots 5 figure Summary) Learning Intention 1. We are learning about Boxplots and five figure summary. 15-May-17 Success Criteria 1. Calculate five figure summary. 2. Be able to construct a boxplot. Created by Mr. Lafferty Maths Dept. Box and Whisker Diagrams. Box plots are useful for comparing two or more sets of data like that shown below for heights of boys and girls in a class. Anatomy of a Box and Whisker Diagram. Lower Lowest Quartile Value Whisker 4 5 Median Upper Quartile Whisker Box 6 7 Highest Value 8 9 10 11 12 Boys 130 140 150 160 170 180 cm Girls Demo 190 Drawing a Box Plot. Example 1: Draw a Box plot for the data below Q2 Q1 4, 4, 5, 6, 8, 8, Lower Quartile = 5½ 4 5 Q3 8, Median = 8 6 7 8 9 9, 9, 9, 10, 12 Upper Quartile = 9 10 11 12 Demo Drawing a Box Plot. Example 2: Draw a Box plot for the data below Q2 Q1 3, 4, 4, 6, 8, Lower Quartile = 4 3 4 5 6 Q3 8, 8, Median = 8 7 8 9 9, 10, 10, 15, Upper Quartile = 10 10 11 12 13 14 15 Demo Drawing a Box Plot. Question: Stuart recorded the heights in cm of boys in his class as shown below. Draw a box plot for this data. Q2 QL Qu 137, 148, 155, 158, 165, 166, 166, 171, 171, 173, 175, 180, 184, 186, 186 Lower Quartile = 158 130 140 Upper Quartile = 180 Median = 171 150 160 170 180 cm Demo 190 Drawing a Box Plot. Question: Gemma recorded the heights in cm of girls in the same class and constructed a box plot from the data. The box plots for both boys and girls are shown below. Use the box plots to choose some correct statements comparing heights of boys and girls in the class. Justify your answers. Boys 130 140 150 160 170 Girls 1. The girls are taller on average. 180 cm 190 Demo 2. The boys are taller on average. 3. The girls height is more consistent. 5. The smallest person is a girl 4. The boys height is more consistent. 6. The tallest person is a boy Statistics www.mathsrevision.com N5 LS Now try N5 TJ Lifeskills Ex 24.4 Ch24 (page 238) 15-May-17 Created by Mr Lafferty Maths Dept Demo Starter Questions www.mathsrevision.com N5 LS In pairs come up with the type of questions you can be asked involving Area and Volume in an exam. 15-May-17 Created by Mr. Lafferty Maths Dept. Standard Deviation For a Sample of Data Standard deviation www.mathsrevision.com N5 LS Learning Intention 1. We are learning how to calculate the Sample Standard deviation for a sample of data. 15-May-17 Success Criteria 1. Know the term Sample Standard Deviation. 2. Calculate the Sample Standard Deviation for a collection of data. Created by Mr. Lafferty Maths Dept. Standard Deviation www.mathsrevision.com N5 LS The range measures spread. Unfortunately any big change in either the largest value or smallest score will mean a big change in the range, even though only one number may have changed. The semi-interquartile range is less sensitive to a single number changing but again it is only really based on two of the score. 15-May-17 Created by Mr. Lafferty Maths Dept. Standard Deviation www.mathsrevision.com N5 LS A measure of spread which uses all the data is the Standard Deviation 15-May-17 Created by Mr. Lafferty Maths Dept. Standard Deviation www.mathsrevision.com N5 LS When Standard Deviation is LOW it means the data values are close to the MEAN. When Standard Deviation is HIGH it means the data values are spread out from the MEAN. Mean 15-May-17 Mean Created by Mr. Lafferty Maths Dept. www.mathsrevision.com N5 LS Standard Deviation For a Sample ofWe Data will use this version because it is easier to use in a sample In real life situations it is normal to work with practice ). ! of data ( survey / questionnaire We can use two formulae to calculate the sample deviation. s ( x x) 2 n 1 s = standard deviation x = sample mean 15-May-17 x 2 s x n 1 ∑ = The sum of n = number in sample Created by Mr. Lafferty Maths Dept. n 2 www.mathsrevision.com 2: Q1a. Calculate the mean : Q1a.Step Calculate the Standard Deviation Step592 1 : ÷ 8 = 74 Step 3 :sample deviation all the values For a SampleSquare of Data find the total N5 LSSum all the values Use formula toand calculate sample have deviation Example 1a : Eight athletes heart rates 70, 72, 73, 74, 75, 76, 76 and 76. s s 15-May-17 2 x x n 1 43842 Heart rate (x) 2 8 1 8 2 4900 72 43842 43808 5184 73 7 s n 592 70 x2 5329 74 5476 75 5625 76 s 4.875 5776 s76 2.2 (to 1 d . p5776 .) 76 Created by Mr. Lafferty Maths Dept. Totals ∑x = 592 5776 ∑x2 = 43842 Q1b(i) Calculate the mean : Standard Deviation Q1b(ii) Calculate the 720 ÷ 8 = 90 sample deviation For a Sample of Data www.mathsrevision.com N5 LS Example 1b : Eight office staff train as athletes. Their Pulse rates are 80, 81, 83, 90, 94, 96, 96 and 100 BPM s s 15-May-17 x 2 x n 1 65218 2 s 81 90 2 94 96 96 7 418 s 7 s 7.7 100 Created by Mr. Lafferty Maths Dept. Totals ∑x = 720 x2 6400 65218 64800 83 720 8 1 80 n 8 Heart rate (x) 6561 6889 8100 8836 9216 9216 (to 1d10000 . p.) ∑x2 = 65218 Standard Q1b(iii) WhoDeviation are fitter Q1b(iv) What does the athletes or of staff. Forthe adeviation Sample Data tell us. Compare means Staff data is more spread Athletes are fitter out. www.mathsrevision.com N5 LS Athletes Staff Mean 74 BPM Mean 90 BPM s 2.2 (to 1d. p.) s 7.7 (to 1d. p.) 15-May-17 Created by Mr. Lafferty Maths Dept. Standard Deviation For a FULL set of Data www.mathsrevision.com N5 LS Now try N5 TJ Lifeskills Ex 24.5 Ch24 (page 241) 15-May-17 Created by Mr. Lafferty Maths Dept. Standard Deviation For a FULL set of Data Have you updated your Learning Log ? www.mathsrevision.com N5 LS Now try N5 TJ Lifeskills Ex 24.5 Ch24 (page 241) Are you on Target ? I can ? 15-May-17 Created by Mr. Lafferty Maths Dept. Are you on Target ? I can ? Mindmap Calculate the mean and standard deviation Go on to next slide for part c Qs b next slide