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Estimate the mean from grouped data workbookWrite your name in this box FORMTEXT ?????__________________________________________________________________________How to use this workbookWhere to write your answersYou will see on each question there are one or more grey boxes. These are for you to type the answer in.Eg: 6 + 4 = FORMTEXT ?????In the box you write your answer.The layout keeps changingOne problem is that the grey boxes are a different size when empty, to when full. This means that questions take up a different amount of space when empty. You will see this means the pages will change as you fill them in. Unfortunately there is nothing I can do about that.Why are you asking me questions about my answers? You will also see that on some questions you are asked to write what has changed from one question to the next. Most questions are in a specific order and only one thing has changed from one question to the next. By looking at what changes from one question to the next, you can see what affect that change has on the answer. That should help you to understand what is going on.You are also asked to say what the affect this had on the answer. Did it double it? Did it increase it by 3, reduce it by a power of x? Once again, by looking at what impact the change had on the answer, it will help you understand what is going on in the question.Grouped frequency tablesSometimes when we're dealing with a lot of data at once it's easier to separate the values into groups (called classes). Then, we can display it in what is called a?grouped frequency table.Frequency tables can be used to log different kinds of data.Discrete data?is information that we collect that can be counted and that only has a certain number of values.?Examples?of?discrete data?include the number of people in a class, test questions answered correctly, goals scored etc.Continuous data?is information that we collect that can be measured. It has an number of values.?Examples?of?continuous data?include height, weight, distance, volume, time taken to complete a task. Within any of these there is an infinite range of values.Example of discrete data frequency tableDiscrete data is counted NumberFrequency0 – 526 – 10511 – 15416 – 20 3Example of continuous data frequency tableContinuous data is measuredNumberFrequency0 ≤ n < 525 ≤ n < 10510 ≤ n < 15415 ≤ n < 20 3Note that in both cases there is only a single class into which a data item could go. For example 5 would go into the first row of the discrete data, but the second row of the continuous. (the first row is values up to but not including 5).It is important that classes must NOT overlap, or have gaps.Correct:0 – 5 0 ≤ n < 56 – 10 5 ≤ n < 10It is clear where 5 goesIncorrect0 – 5 0 < n < 55 – 10 5 < n < 10Where would 5 go?75256533573300Mathswatch 1: Discrete dataA rugby team scored the following number of points in its last 20 matches. Record this data in a suitable frequency table0, 5, 13, 10, 7, 7, 5, 23, 14, 18, 15, 24, 0, 3, 17, 15, 27, 8, 10, 0Step 1: Discrete or continuous data?Points are counted, so are discrete dataStep 2: Design a table. Typically, we want around five or so classes, or groups. Minimum value: 0Maximum value : 27PointsFrequency0 – 566 – 10511 – 15416 – 20 221 – 25226 – 301Example 2: Continuous dataThe weight, in kg, of a certain product was as follows. Record this data in a suitable frequency table5.00, 4.21, 4.97, 5.00, 5.00, 5.29, 5.05, 5.50, 5.74,5.00, 5.40, 5.20, 5.10, 5.06, 4.50, 4.50, 5.50, 4.50, 5.00, 5.00, 5.50, 4.82Step 1: Discrete or continuous data?Weights are measured, so are discrete dataStep 2: Design a table. Typically, we want around six or so classes, or groups. Minimum value: 4.21Maximum value : 5.74Weight (w)Frequency4.00 ≤ n < 4.2514.25 ≤ n < 4.5004.50 ≤ n < 4.7534.75 ≤ n < 5.0025.00 ≤ n < 5.25105.25 ≤ n < 5.5025.50 ≤ n < 5.754Record this data in a suitable frequency tableQ1.The following are the heights of trees in a garden in metres. Record this data in a suitable frequency table1.41, 1.43, 1.58, 1.62, 1.65, 1.80, 1.82, 1.82, 1.86, 1.98, 2.12, 2.16, 2.25, 2.29, 2.45, 2.46, 2.50, 2.50, 2.54, 2.58Step 1: Discrete or continuous data?Heights are measured, so are continuous dataMinimum value: 1.41mMaximum value : 2.58mHeight (m)Frequency1.25 ≤ h < 1.50 FORMTEXT ?????1.50 ≤ h < 1.75 FORMTEXT ?????1.75 ≤ h < 2.00 FORMTEXT ?????2.00 ≤ h < 2.25 FORMTEXT ?????2.25 ≤ h < 2.50 FORMTEXT ?????2.50 ≤ h < 2.75 FORMTEXT ?????Q2The following are the points scored in a recent test. Record this data in a suitable frequency table10, 10, 12, 13, 14, 14, 17, 17, 18, 20, 20, 21, 21, 24, 27, 27, 27, 31, 32, 33, 35, 36, 36, 37, 38Step 1: Discrete or continuous data?Points are counted, so are FORMTEXT ????? dataMinimum value: FORMTEXT ?????Maximum value : FORMTEXT ?????Points Frequency10 – 15 FORMTEXT ?????16 – 20 FORMTEXT ?????21 – 25 FORMTEXT ????? FORMTEXT ????? – FORMTEXT ????? FORMTEXT ????? FORMTEXT ????? – FORMTEXT ????? FORMTEXT ????? FORMTEXT ????? – FORMTEXT ????? FORMTEXT ?????Q3 The following are the times of 100m races in seconds. Record this data in a suitable frequency table9.15, 9.25, 9.25, 9.34, 9.35, 9.37, 9.37, 9.49, 9.55, 9.58, 9.66, 9.69, 9.74, 9.84, 9.85, 9.85, 9.91, 10.00, 10.01, 10.06, 10.17, 10.18, 10.30, 10.31, 10.33Step 1: Discrete or continuous data?Time is FORMTEXT ?????, so is FORMTEXT ?????dataMinimum value: FORMTEXT ?????Maximum value FORMTEXT ?????Time (s)Frequency FORMTEXT ?????≤ t < FORMTEXT ????? FORMTEXT ????? FORMTEXT ?????≤ t < FORMTEXT ????? FORMTEXT ????? FORMTEXT ?????≤ t < FORMTEXT ????? FORMTEXT ????? FORMTEXT ?????≤ t < FORMTEXT ????? FORMTEXT ????? FORMTEXT ?????≤ t < FORMTEXT ????? FORMTEXT ????? FORMTEXT ?????≤ t < FORMTEXT ????? FORMTEXT ????? FORMTEXT ?????≤ t < FORMTEXT ????? FORMTEXT ????? FORMTEXT ?????≤ t < FORMTEXT ????? FORMTEXT ????? FORMTEXT ?????≤ t < FORMTEXT ????? FORMTEXT ????? FORMTEXT ?????≤ t < FORMTEXT ????? FORMTEXT ?????(note: design your own table using however many classes you think necessary. You do NOT have to fill every row of the table.)Calculating the midpoint of two numbersBefore working out how to estimate the mean we need to be able to work out the midpoint of two numbers.The midpoint is the number that lies half way between two values.MP = first number+second number2Example 1: Two integersCalculate the midpoint between 8 and 12Step 1: Add the two numbers8+12=20Step 2: Divide by two202=10The midpoint of 8 and 12 is 10.Example 2: Two decimalsCalculate the midpoint between 8.5 and 12.6Step 1: Add the two numbers8.5+12.6=21.1Step 2: Divide by two21.12=10.55The midpoint of 8.5 and 12.6 is 10.55Calculate the midpoint for the following tablesQ1. 8 and 10Step 1: 8 + 10 = 18Step 2 18/2 = FORMTEXT ?????The midpoint of 8 and 10 is FORMTEXT ?????Q27 and 11Step 1: 7 + 11 = FORMTEXT ?????Step 2 18/2 = FORMTEXT ?????The midpoint of 7 and 11 is FORMTEXT ?????What changed between Q1 and Q2 FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q32 and 16Step 1: 2 + 16 = FORMTEXT ?????Step 2 FORMTEXT ?????/2 = FORMTEXT ?????The midpoint of 2 and 16 is FORMTEXT ?????What changed between Q2 and Q3 FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q422 and 36Step 1: FORMTEXT ????? + FORMTEXT ?????= FORMTEXT ?????Step 2 FORMTEXT ?????/2 = FORMTEXT ?????The midpoint of 22 and 36is FORMTEXT ?????What changed between Q3 and Q4? FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q522 and 46Step 1: FORMTEXT ????? + FORMTEXT ?????= FORMTEXT ?????Step 2 FORMTEXT ?????/2 = FORMTEXT ?????The midpoint of 22 and 46is FORMTEXT ?????What changed between Q4 and Q5 FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q622 and 47Step 1: FORMTEXT ????? + FORMTEXT ?????= FORMTEXT ?????Step 2 FORMTEXT ?????/2 = FORMTEXT ?????The midpoint of 22 and 47is FORMTEXT ?????What changed between Q5 and Q6 FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q722 and 48Step 1: FORMTEXT ????? + FORMTEXT ?????= FORMTEXT ?????Step 2 FORMTEXT ?????/2 = FORMTEXT ?????The midpoint of 22 and 48is FORMTEXT ?????What has changed between Q6 and Q7? FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q821 and 48Step 1: FORMTEXT ????? + FORMTEXT ?????= FORMTEXT ?????Step 2 FORMTEXT ?????/2 = FORMTEXT ?????The midpoint of 21 and 48is FORMTEXT ?????What has changed between Q7 and Q8? FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q921 and 47Step 1: FORMTEXT ????? + FORMTEXT ?????= FORMTEXT ?????Step 2 FORMTEXT ?????/2 = FORMTEXT ?????The midpoint of 21 and 47is FORMTEXT ?????What has changed between Q8 and Q9? FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q1042 and 94Step 1: FORMTEXT ????? + FORMTEXT ?????= FORMTEXT ?????Step 2 FORMTEXT ?????/2 = FORMTEXT ?????The midpoint of 42 and 94is FORMTEXT ?????What has changed between Q9 and Q10? FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q11142 and 194Step 1: FORMTEXT ????? + FORMTEXT ?????= FORMTEXT ?????Step 2 FORMTEXT ?????/2 = FORMTEXT ?????The midpoint of 142 and 194is FORMTEXT ?????What has changed between Q10 and Q11? FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q1214.2 and 19.4Step 1: FORMTEXT ????? + FORMTEXT ?????= FORMTEXT ?????Step 2 FORMTEXT ?????/2 = FORMTEXT ?????The midpoint of 14.2 and 19.4is FORMTEXT ?????What has changed between Q11 and Q12? FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q137.1 and 9.7Step 1: FORMTEXT ????? + FORMTEXT ?????= FORMTEXT ?????Step 2 FORMTEXT ?????/2 = FORMTEXT ?????The midpoint of 7.1 and 9.7is FORMTEXT ?????What has changed between Q12 and Q13? FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q147 and 9.6Step 1: FORMTEXT ????? + FORMTEXT ?????= FORMTEXT ?????Step 2 FORMTEXT ?????/2 = FORMTEXT ?????The midpoint of 7 and 9.6is FORMTEXT ?????What has changed between Q13 and Q14? FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q15-9.6 and -7Step 1: FORMTEXT ????? + FORMTEXT ?????= FORMTEXT ?????Step 2 FORMTEXT ?????/2 = FORMTEXT ?????The midpoint of -9.6 and -7is FORMTEXT ?????What has changed between Q14 and Q14? FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Estimating the mean from grouped dataRecall how you found the mean of table dataPointsFrequencyp x f5156318742882161067Mean = Total points Total frequency= 6710= 6.7With grouped data, what figure do we use for the points? The bottom of the group (6, 11, 16 etc)? The top of the group (10, 15, 20 etc)? Both would cause problems.PointsFrequency6 - 10111 - 15316 - 20421 - 25226 - 30111With grouped data we use the midpoint of the class, and multiply that by the frequency. You need to add two extra columns.PointsFrequencympmp x f6 - 101(6 +10)/2 = 88 × 1 = 811 - 153(11 +15)/2 = 1313 × 3 = 3916 - 204(16 +20)/2 = 1818 × 4 = 7221 - 252(21 +25)/2 = 2323 × 2 = 4626 - 301(26 +30)/2 = 2828 × 1 = 2811193Mean = Total points Total frequency= 19311= 17.5It is the same for continuous dataHeightFrequencympmp x f5 < h ≤ 101(5 +10)/2 = 7.57.5 × 1 = 7.510 < h ≤ 153(10 +15)/2 = 12.512.5 × 3 = 37.515 < h ≤ 204(15 +20)/2 = 17.517.5 × 4 = 7020 < h ≤ 252(20 +25)/2 = 22.522.5 × 2 = 4525 < h ≤ 301(25 +30)/2 = 27.527.5 × 1 = 27.511187.5Mean = Total points Total frequency= 187.511= 17.03575413-2476500Mathswatch Estimate the mean from the following tablesQ1Mass, x (kg)FrequencyMPMP × f0<x≤1010 + 102=55 × 1 = 510<x≤20210 + 202=1515 × 2 = 3020<x≤30420 + 302=2525 × 4 = 10030<x≤40330 + 402=3535 × 3 = 10510240Mean = Total mass Total frequency= FORMTEXT ????? FORMTEXT ?????= FORMTEXT ?????Q2Mass, x (kg)FrequencyMPMP × f0<x≤1020 + 102=55 × 2 = 1010<x≤20410 + 202=1515 × FORMTEXT ????? = FORMTEXT ?????20<x≤30820 + 302=2525 × FORMTEXT ????? = FORMTEXT ?????30<x≤40630 + 402=3535 × FORMTEXT ????? = FORMTEXT ?????20480Mean = Total mass Total frequency= FORMTEXT ????? FORMTEXT ?????= FORMTEXT ?????What has changed between Q1 and Q2? FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q3Mass, x (kg)FrequencyMPMP × f0<x≤2020 + FORMTEXT ?????2=1010 × 2 = 2020<x≤404 FORMTEXT ????? + FORMTEXT ?????2=3030 × FORMTEXT ????? = FORMTEXT ?????40<x≤608 FORMTEXT ????? + FORMTEXT ?????2=5050 × FORMTEXT ????? = FORMTEXT ?????60<x≤806 FORMTEXT ????? + FORMTEXT ?????2=7070 × FORMTEXT ????? = FORMTEXT ?????20 FORMTEXT ?????Mean = Total mass Total frequency= FORMTEXT ????? FORMTEXT ?????= FORMTEXT ?????What has changed between Q2 and Q3? FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q4Mass, x (kg)FrequencyMPMP × f0<x≤206 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × 6 = FORMTEXT ?????20<x≤408 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????40<x≤604 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????60<x≤802 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????20 FORMTEXT ?????Mean = Total mass Total frequency= FORMTEXT ????? FORMTEXT ?????= FORMTEXT ?????What has changed between Q3 and Q4? FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q5Mass, x (kg)FrequencyMPMP × f0<x≤206 FORMTEXT ?????+ FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????20<x≤408 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????40<x≤604 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????60<x≤8020 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ????? FORMTEXT ????? FORMTEXT ?????Mean = Total mass Total frequency= FORMTEXT ????? FORMTEXT ?????= FORMTEXT ?????What has changed between Q4 and Q5? FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q6Mass, x (kg)FrequencyMPMP × f0<x≤206 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????20<x≤408 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????40<x≤6020 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????60<x≤8020 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ????? FORMTEXT ????? FORMTEXT ?????Mean = Total mass Total frequency= FORMTEXT ????? FORMTEXT ?????= FORMTEXT ?????What has changed between Q5 and Q6? FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q7Mass, x (kg)FrequencyMPMP × f0<x≤206 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????20<x≤408 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????40<x≤600 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????60<x≤8020 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ????? FORMTEXT ????? FORMTEXT ?????Mean = Total mass Total frequency= FORMTEXT ????? FORMTEXT ?????= FORMTEXT ?????What has changed between Q6 and Q7? FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q8Mass, x (kg)FrequencyMPMP × f1<x≤216 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????21<x≤418 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????41<x≤610 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????61<x≤8120 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ????? FORMTEXT ????? FORMTEXT ?????Mean = Total mass Total frequency= FORMTEXT ????? FORMTEXT ?????= FORMTEXT ?????What has changed between Q7 and Q8? FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q9Mass, x (kg)FrequencyMPMP × f1<x≤216 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????21<x≤418 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????41<x≤610 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????61<x≤665 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????66<x≤8115 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ????? FORMTEXT ????? FORMTEXT ?????Mean = Total mass Total frequency= FORMTEXT ????? FORMTEXT ?????= FORMTEXT ?????What has changed between Q8 and Q9? FORMTEXT ?????How did this affect the answer? FORMTEXT ?????Q10Mass, x (kg)FrequencyMPMP × f1<x≤216 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????21<x≤418 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ?????41<x≤8120 FORMTEXT ????? + FORMTEXT ?????2= FORMTEXT ????? FORMTEXT ????? × FORMTEXT ????? = FORMTEXT ????? FORMTEXT ????? FORMTEXT ?????Mean = Total mass Total frequency= FORMTEXT ????? FORMTEXT ?????= FORMTEXT ?????What has changed between Q9 and Q10? FORMTEXT ?????How did this affect the answer? FORMTEXT ????? ................
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