GCSE Maths Starter 16 Round 0.0536 to 2 significant figures 18÷3 +(20÷5) Factorise 6a + 10 The mean of five numbers is 8. Four of the numbers are 7, 11, 12 and 4. What is the fifth number? Copy this pattern into your book. Next shade one more square so the pattern has a rotational symmetry of order two.

Lesson 16 Frequency polygons (Cumulative frequency H) Mathswatch clip (88[151/152]). To draw a frequency polygon (Grade D ) To draw a cumulative frequency curve (Grade B) To draw a box plot (Grade B)

Foundation

Types of Data Discrete – can only take specific values, e.g. siblings, key stage 3 levels, numbers of objects Continuous Data – can take any value, e.g. height, weight, age, time, etc.

Midpoints Green Red 0 – 10 54 – 56 40 – 50 5 – 9 30 – 40 8 – 14 What is the midpoint between the following numbers? Green 0 – 10 40 – 50 30 – 40 0 – 100 0 – 50 Red 54 – 56 5 – 9 8 – 14 38 – 52 0 – 5

Answers What is the midpoint between the following numbers? Green 5 45 35 50 25 Red 55 7 11 45 2.5

Test Scores 22 12 17 27.5 Frequency 7 Marks A frequency polygon can be drawn directly from the frequency table by finding the mid-point of each class interval. Marks 5 – 9 10 – 14 15 – 19 20 – 24 25 - 30 frequency 4 10 20 13 8 Test Scores 20 17 22 12 15 27.5 Frequency 10 7 5 5-9 10-14 15-19 20-24 25-30 Marks

Time Taken for Race Frequency Time in minutes A frequency polygon can be drawn directly from the frequency table by using by finding the mid-point of each class interval. Time 10 – 20 20 – 30 30 – 40 40 – 50 50 - 60 frequency 7 10 18 6 4 Time Taken for Race 20 15 Frequency 10 5 10 20 30 40 50 60 Time in minutes

Lesson 16 Frequency polygons (Cumulative frequency H) Mathswatch clip (88[151/152]). Foundation

Lesson 16 Frequency polygons (Cumulative frequency H) Mathswatch clip (88[151/152]). Exam questions

Lesson 16 Frequency polygons (Cumulative frequency H) Mathswatch clip (88[151/152]). Exam questions

Higher

Cumulative Frequency Curves Remember: When data is grouped we don’t know the actual value of either the mean, median, mode or range. We can get an estimate for the mean by using mid-points from the frequency table. midpoint(x) mp x f 2 50 - 60 4 40 - 50 5 30 - 40 7 20 - 30 10 10 - 20 27 0 - 10 frequency minutes late The measure of spread used with the median is the inter- quartile range. This is a better measure of spread as it only uses the middle half of the data that is grouped around the median. This means that unlike the range it is not subject to extreme values. Remember: The measure of spread used with the mean is the range. The range is not a good measure of spread as it is subject to extreme values. We can also use the grouped data to obtain an estimate of the median and a measure of spread called the inter-quartile range. We do this by plotting a cumulative frequency curve (Ogive).

Discuss the calculations below Cumulative Frequency Curves Remember: When data is grouped we don’t know the actual value of either the mean, median, mode or range. We can get an estimate for the mean by using mid-points from the frequency table. midpoint(x) mp x f 2 50 - 60 4 40 - 50 5 30 - 40 7 20 - 30 10 10 - 20 27 0 - 10 frequency minutes late 2, 5, 6, 6, 7, 8, 8, 8, 9, 9, 10, 15 Median = 8 hours and the inter-quartile range = 9 – 6 = 3 hours. Battery Life: The life of 12 batteries recorded in hours is: Mean = 93/12 = 7.75 hours and the range = 15 – 2 = 13 hours. Discuss the calculations below

Cumulative Frequency Curves Cumulative frequency diagrams are used to obtain an estimate of the median, and quartiles. from a set of grouped data. Constructing a cumulative frequency table is first step. Cumulative Frequency Curves Cumulative frequency table < 60 5 50 - 60 < 50 8 40 - 50 < 40 12 30 - 40 < 30 22 20 - 30 < 20 10 - 20 < 10 0 - 10 Cumulative Frequency Upper Limit Frequency Minutes Late Example 1. During a 4 hour period at a busy airport the number of late-arriving aircraft was recorded. 5 13 35 47 55 60 Cumulative frequency just means running total.

Cumulative frequency table 60 < 60 5 50 - 60 55 < 50 8 40 - 50 47 < 40 12 30 - 40 35 < 30 22 20 - 30 13 < 20 10 - 20 < 10 0 - 10 CF Upper Limit f Mins Late 10 20 30 40 50 70 Cumulative Frequency Minutes Late Plotting the curve ¾ UQ = 38 ½ IQR = 38 – 21 = 17 mins Median = 27 ¼ LQ = 21 Plot the end point of each interval against cumulative frequency, then join the points to make the curve. Get an estimate for the median. Find the lower quartile. Find the Upper Quartile. Find the Inter Quartile Range.(IQR = UQ - LQ)

Cumulative Frequency Curves Cumulative frequency diagrams are used to obtain an estimate of the median and quartiles from a set of grouped data. Constructing a cumulative frequency table is first step. Cumulative Frequency Curves Cumulative frequency table Example 2. A P.E teacher records the distance jumped by each of 70 pupils. Distance (cm) No of pupils Upper Limit Cumulative Frequency 180  d  190 2 d  190 2 190  d  200 6 d  200 8 200  d  210 9 d  210 17 210  d  220 7 d  220 24 220  d  230 15 d  230 39 230  d  240 18 d  240 57 240  d  250 8 d  250 65 250  d  260 5 d  260 70 Cumulative frequency just means running total.

Cumulative Frequency Table 10 20 30 40 50 60 70 180 190 200 210 220 230 240 250 260 Cumulative Frequency Distance jumped (cm) 5 250  d  260 65 8 240  d  250 57 18 230  d  240 39 15 220  d  230 24 7 210  d  220 17 9 200  d  210 6 190  d  200 2 180  d  190 Number of pupils Plotting The Curve Cumulative Frequency Table ¾ UQ = 237 ½ IQR = 237 – 212 = 25 cm Median = 227 ¼ LQ= 212 Plot the end point of each interval against cumulative frequency, then join the points to make the curve. Find the Lower Quartile. Get an estimate for the median. Find the Upper Quartile. Find the Inter Quartile Range.(IQR = UQ - LQ)

Interpreting Cumulative Frequency Curves 10 20 30 40 50 60 70 Cumulative Frequency Minutes Late Interpreting Cumulative Frequency Curves The cumulative frequency curve gives information on aircraft arriving late at an airport. Use the curve to find estimates to (a) The median (b) The inter-quartile range (c) The number of aircraft arriving less than 45 minutes late. (d) The number of aircraft arriving more than 25 minutes late. ¾ UQ =38 ½ IQR = 38 – 21 = 17 mins Median = 27 ¼ LQ = 21

Interpreting Cumulative Frequency Curves 10 20 30 40 50 60 70 Cumulative Frequency Minutes Late Interpreting Cumulative Frequency Curves The cumulative frequency curve gives information on aircraft arriving late at an airport. Use the curve to find estimates to: (a) The median (b) The inter-quartile range (c) The number of aircraft arriving less than 45 minutes late. (d) The number of aircraft arriving more than 25 minutes late. 52 60 – 24 =36

Interpreting Cumulative Frequency Curves 10 20 30 40 50 60 70 Cumulative Frequency Marks The graph shows the cumulative frequency curve of the marks for students in an examination. Use the graph to find: (a) The median mark. (b) The number of students who got less than 55 marks. (c) The pass mark if ¾ of the students passed the test. 58 Median = 27 21 ¾ of the students passing the test implies that ¼ failed. (15 students)

Interpreting Cumulative Frequency Curves 20 40 60 80 100 120 140 Cumulative Frequency 200 300 400 500 600 Lifetime of bulbs in hours The lifetime of 120 projector bulbs was measured in a laboratory. The graph shows the cumulative frequency curve for the results. Use the graph to find: (a) The median lifetime of a bulb. (b) The number of bulbs that had a lifetime of between 200 and 400 hours? (c) After how many hours were 80% of the bulbs dead?. (d) What was the shortest lifetime of a bulb? (a) 330 hours (b) 86 - 12 = 74 (c) 440 hours (d) 100 hours

Box Plot from Cumulative Frequency Curve 10 20 30 40 50 60 70 Cumulative Frequency Minutes Late Median = 27 LQ = 21 UQ = 38 IQR = 38 – 21 = 17 mins ½ ¼ ¾ 10 20 30 40 50 60

Lesson 16 Frequency polygons (Cumulative frequency H) Mathswatch clip (88[151/152]). < 60 5 50 - 60 < 50 8 40 - 50 < 40 12 30 - 40 < 30 22 20 - 30 < 20 10 - 20 < 10 0 - 10 CF Upper Limit f Mins Late 10 20 30 40 50 60 70 Cumulative Frequency Minutes Late

Lesson 16 Frequency polygons (Cumulative frequency H) Mathswatch clip (88[151/152]). Exam questions

Lesson 16 Frequency polygons (Cumulative frequency H) Mathswatch clip (88[151/152]). Exam questions

Lesson 16 Frequency polygons (Cumulative frequency H) Mathswatch clip (88[151/152]). Exam questions

Cumulative Frequency Minutes Late 70 60 50 40 30 20 10 < 60 5 50 - 60 < 50 8 40 - 50 < 40 12 30 - 40 < 30 22 20 - 30 < 20 10 - 20 < 10 0 - 10 CF Upper Limit f Mins Late 10 20 30 40 50 60 70 Cumulative Frequency Minutes Late