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© Christine Crisp “Teach A Level Maths” Statistics 1.

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1 © Christine Crisp “Teach A Level Maths” Statistics 1

2 Introduction to S1 You met some statistical diagrams when you did GCSE. The next three presentations and this one remind you of them and point out some details that you may not have met before. We will start with stem and leaf diagrams ( including back-to-back ). Stem and leaf diagrams are sometimes called stem plots.

3 Introduction to S1 Weekly hours of 30 men e.g. The table below gives the number of hours worked in a particular week by a sample of 30 men The stem shows the tens... I’ll use intervals of 5 hours to draw the diagram i.e , etc. and the leaves the units e.g. 46 is 4 tens and 6 units

4 Introduction to S1 Weekly hours of 30 men e.g. The table below gives the number of hours worked in a particular week by a sample of 30 men I’ll use intervals of 5 hours to draw the diagram i.e , etc e.g. 46 is 4 tens and 6 units Weekly hours of 30 men The stem shows the tens... and the leaves the units

5 Introduction to S1 Weekly hours of 30 men e.g. The table below gives the number of hours worked in a particular week by a sample of 30 men I’ll use intervals of 5 hours to draw the diagram i.e , etc e.g. 46 is 4 tens and 6 units Weekly hours of 30 men N.B. 35 goes here... not in the line below. The stem shows the tens... and the leaves the units

6 Introduction to S1 e.g. The table below gives the number of hours worked in a particular week by a sample of 30 men I’ll use intervals of 5 hours to draw the diagram i.e , etc e.g. 46 is 4 tens and 6 units We must show a key. Key: 3 5 means 35 hours Weekly hours of 30 men The stem shows the tens... and the leaves the units

7 Introduction to S Weekly hours of 30 men If you tip your head to the right and look at the diagram you can see it is just a bar chart with more detail. Points to notice: The leaves are in numerical order The diagram uses raw ( not grouped ) data Key: 3 5 means 35 hours

8 Introduction to S1 The data below is a back to back stem and leaf diagram giving the weight in grams of eggs collected from ostriches and emus. This method can be used to compare two sets of data. Ostrich Emu Key 27|2 = 272

9 Introduction to S1 0–402 5– – – – – –343 35–3936 A grouped data stem and leaf diagram Data 2, 5, 5, 8, 12, 16, 17, 17, 19, 20, 22, 22, 24, 25, 25, 25, 27, 27, 29, 29, 36 Draw a stem and leaf diagram using groupings 0–4, 5–9, 10–14 etc Key 1/2 = 12 3/6 = 36

10 Introduction to S1 Histogram : A bar chart with continuous data. The bars are drawn up to the class boundaries. NO GAPS between bars. The class boundary occurs halfway between the boundaries of two successive groups. (Except in age questions) Groups 0-9, 10-19, etc. the class boundaries between each group occur at 9.5, 19.5 So any quantity >9.5 is in group 2 and any quantity <9.5 is in group 1. The bars are drawn at 9.5 and 19.5 etc. It is very important that the area under each bar is proportional to the frequency.

11 Introduction to S1 Suppose the data are grouped so that those below 20 and above 69 are combined. e.g. The projected population of the U.K. for 2005 ( by age ) Source: USA IDB – – – – – – – – – 9 (millions)( years ) FreqAGE Histograms

12 Introduction to S1 e.g. The projected population of the U.K. for 2005 ( by age ) Source: USA IDB – – – – – – – – – 9 (millions)( years ) FreqAGE Suppose the data are grouped so that those below 20 and above 69 are combined AGE (years) Freq (millions) To draw the diagram we must have an upper class value

13 Introduction to S1 e.g. The projected population of the U.K. for 2005 ( by age ) Source: USA IDB Suppose the data are grouped so that those below 20 and above 69 are combined. I chose a sensible figure Freq (millions) AGE (years) Source: USA IDB – – – – – – – – – 9 (millions)( years ) FreqAGE

14 Introduction to S1 e.g. The projected population of the U.K. for 2005 ( by age ) Freq (millions) AGE (years) If we use the data below to draw an age/frequency graph then it is very misleading as the 1 st and last bar dominate So frequencies are represented by areas Bar1 1 should represent just over twice as many people as bar 2 but it appears to be about 4 times as many

15 Introduction to S1 A histogram shows frequencies as areas. To draw the histogram, we need to find the width and height of each column. The width is the class width: upper class boundary (u.c.b.) minus lower class boundary (l.c.b.) Freq (millions) AGE (years) Class width 20 Since these are ages, the 1 st class, for example, has u.c.b. = 20 and the l.c.b. = 0, so the width is 20.

16 Introduction to S1 A histogram shows frequencies as areas. e.g. The projected population of the U.K. for 2005 ( by age )  height = frequency width The width is the class width: upper class boundary (u.c.b.) minus lower class boundary (l.c.b.). Area of a rectangle = width  height To draw the histogram, we need to find the width and height of each column. So, frequency = width  height Class width Freq (millions) AGE (years) 40 10

17 Introduction to S Class width Freq (millions) AGE (years) Freq density A histogram shows frequencies as areas. e.g. The projected population of the U.K. for 2005 ( by age ) The height is called the frequency density The width is the class width: upper class boundary (u.c.b.) minus lower class boundary (l.c.b.). e.g. For the 1 st class, freq. density = To draw the histogram, we need to find the width and height of each column. height = frequency width

18 Introduction to S Freq density Class width Freq (millions) AGE (years) A histogram shows frequencies as areas. e.g. The projected population of the U.K. for 2005 ( by age ) The height is called the frequency density The width is the class width: upper class boundary (u.c.b.) minus lower class boundary (l.c.b.). e.g. For the 1 st class, freq. density = To draw the histogram, we need to find the width and height of each column. height = frequency width 0 ·75

19 Introduction to S Freq density Class width Freq (millions) AGE (years) A histogram shows frequencies as areas. e.g. The projected population of the U.K. for 2005 ( by age ) The width is the class width: upper class boundary (u.c.b.) minus lower class boundary (l.c.b.). We can now draw the histogram. To draw the histogram, we need to find the width and height of each column. The height is called the frequency density  height = frequency width 0 ·75 0 ·15 0 ·6 0 ·8 0 ·9 0 ·7

20 Introduction to S1 AGE (years) Freq (millions) Class width Freq density · · · · · · ·15 The projected population of the U.K. for 2005 ( by age ) Notice that the frequencies for the last 2 classes are the same. On the histogram the areas showing these classes are the same. If we had plotted frequency on the y -axis, the diagram would be very misleading. ( It would suggest there are 6 million in each age group 70 – 79, 80 – 89, 90 – 99 and 100 – 109. )

21 Introduction to S1 SUMMARY  Frequency is shown by area.  The y -axis is used for frequency density. Histograms are used to display grouped frequency data.  Class width is given by u.c.b. – l.c.b. where, u.c.b. is upper class boundary and l.c.b. is lower class boundary  frequency density =

22 Introduction to S1 Exercise 95 components are tested until they fail. The table gives the times taken ( hours ) until failure. Time to failure (hours) Number of components Find 3 things wrong with the histogram which represents the data in the table.

23 Introduction to S1 Answer: Time to failure (hours) Number of components Frequency has been plotted instead of frequency density. There is no title. There are no units on the x -axis.

24 Introduction to S1 Time taken for 95 components to fail Incorrect diagram Correct diagram

25 Introduction to S1 Length of millipedeClass boundaries FrequencyClass widthFreq. Density 0 – 90 – – – – – Note Bars drawn at 9.5, 19.5 and 39. Freq density length Histogram showing length of millipede

26 Introduction to S1 Source: USA IDB – – – – – – – – – 9 (millions) ( years ) Cu.FFreqAGE ANS: The data are given to the nearest million. The projected figure was 113,000. Why does this appear as 0? In drawing the diagram I shall miss out this group. e.g. The projected population of the U.K. for 2005, by age: Cumulative Frequency Graphs

27 Introduction to S1 Source: USA IDB – – – – – – – – – 9 (millions) ( years ) Cu.FFreqAGE Points are plotted at upper class boundaries (u.c.bs.) Points to notice: e.g. the u.c.b. for 0  9 would normally be 9·5 There is no gap between 9 and 10 as the data are continuous. e.g. The projected population of the U.K. for 2005, by age:

28 Introduction to S1 Source: USA IDB – – – – – – – – – 9 (millions) ( years ) Cu.FFreqAGE Points to notice: e.g. the u.c.b. for 0  9 would normally be 9·5 Age data have different u.c.bs. Can you say why this is? ANS: If I ask children their ages, they reply 9 even if they are nearly 10, so, the 0-9 group contains children right up to age 10 NOT just nine and a half. Points are plotted at upper class boundaries (u.c.bs.) There is no gap between 9 and 10 as the data are continuous. e.g. The projected population of the U.K. for 2005, by age:

29 Introduction to S1 e.g. The projected population of the U.K. for 2005, by age: Points to notice: The u.c.bs. for this data set are 10, 20, 30,... Source: USA IDB – – – – – – – – – 9 (millions) ( years ) Cu.FFreqAGE e.g. the u.c.b. for 0  9 would normally be 9·5 Points are plotted at upper class boundaries (u.c.bs.) There is no gap between 9 and 10 as the data are continuous.

30 Introduction to S1 The projected population of the U.K. for 2005 ( by age ) Age (yrs) The median age is estimated as the age corresponding to a cumulative frequency of 30 million. The median age is 39 years ( Half the population of the U.K. will be over 39 in ) e.g. The projected population of the U.K. for 2005, by age: Source: USA IDB – – – – – – – – – 9 ( yrs )(m) ( yrs ) u.c.b.Cu.ffAGE

31 Introduction to S1 The projected population of the U.K. for 2005 ( by age ) Age (yrs) The quartiles are found similarly: lower quartile: 20 years upper quartile: 56 years e.g. The projected population of the U.K. for 2005, by age: The projected population of the U.K. for 2005 ( by age ) Source: USA IDB – – – – – – – – – 9 ( yrs )(m) ( yrs ) u.c.b.Cu.ffAGE The interquartile range is 36 years LQ =  (n+1)th item of data UQ =  (n+1)th item of data

32 Introduction to S1 The projected population of the U.K. for 2005 ( by age ) Age (yrs) If the retirement age were to be 65 for everyone, how many people would be retired? ANS: ( 60 – 51 ) million = 9 million e.g. The projected population of the U.K. for 2005, by age: 51 Source: USA IDB – – – – – – – – – 9 ( yrs )(m) ( yrs ) u.c.b.Cu.ffAGE

33 Introduction to S1 Exercise The table and diagram show the number of flowers in a sample of 43 antirrhinum plants Cu.ffx Source: O.N.Bishop Number of flowers on antirrhinum plants Estimate the median number of plants and the percentage of plants that have more than 90 flowers.

34 Introduction to S1 Number of flowers on antirrhinum plants Cu.ffx The u.c.bs. ( where we plot the points ) are at 39·5, 59·5 etc. Solution: Number with more than 90 flowers = There are 43 observations, so the median is given by the 21·5 th one. Median = 70 32

35 Introduction to S1 Number of flowers on antirrhinum plants Cu.ffx Solution: 32 There are 43 observations, so the median is given by the 21·5 th one. Number with more than 90 flowers = Median = 70 Percentage with more than 90 flowers 43 – 32 = 11 26%

36 Introduction to S1

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