Teach A Level Statistics Maths 1 Stem and

“Teach A Level Statistics Maths” 1 Stem and Leaf Diagrams © Christine Crisp

Stem and Leaf Diagrams Statistics 1 AQA EDEXCEL MEI/OCR "Certain images and/or photos on this presentation are the copyrighted property of Jupiter. Images and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from Jupiter. Images"

Stem and Leaf Diagrams 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.

Stem and Leaf Diagrams e. g. The table below gives the number of hours worked in a particular week by a sample of 30 men 35 41 33 31 30 45 35 36 51 32 32 30 28 35 34 33 35 36 32 42 33 31 41 21 34 35 34 46 32 35 I’ll use intervals of 5 hours to draw the diagram i. e. 20 -25, 26 -30 etc. 5 1 Weekly hours of 30 men The stem shows the 4 5 6 tens. . . and the 4 1 1 2 leaves the units 3 5 5 5 6 6 e. g. 46 is 4 tens and 6 3 0 0 1 1 2 2 3 3 3 4 4 4 units 2 8 2 1

Stem and Leaf Diagrams e. g. The table below gives the number of hours worked in a particular week by a sample of 30 men 35 41 33 31 30 45 35 36 51 32 32 30 28 35 34 33 35 36 32 42 33 31 41 21 34 35 34 46 32 35 I’ll use intervals of 5 hours to draw the diagram i. e. 20 -25, 26 -30 etc. 5 1 Weekly hours of 30 men The stem shows the 4 5 6 tens. . . and the 4 1 1 2 leaves the units 3 5 5 5 6 6 e. g. 46 is 4 tens and 6 3 0 0 1 1 2 2 3 3 3 4 4 4 units 2 8 2 1

Stem and Leaf Diagrams e. g. The table below gives the number of hours worked in a particular week by a sample of 30 men 35 41 33 31 30 45 35 36 51 32 32 30 28 35 34 33 35 36 32 42 33 31 41 21 34 35 34 46 32 35 I’ll use intervals of 5 hours to draw the diagram i. e. 20 -25, 26 -30 etc. 5 1 Weekly hours of 30 men The stem shows the 4 5 6 tens. . . and the 4 1 1 2 leaves the units 3 5 5 5 6 6 e. g. 46 is 4 tens and 6 3 0 0 1 1 2 2 3 3 3 4 4 4 units 2 8 N. B. 35 goes here. . . 2 1 not in the line below.

Stem and Leaf Diagrams e. g. The table below gives the number of hours worked in a particular week by a sample of 30 men 35 41 33 31 30 45 35 36 51 32 32 30 28 35 34 33 35 36 32 42 33 31 41 21 34 35 34 46 32 35 I’ll use intervals of 5 hours to draw the diagram i. e. 20 -25, 26 -30 etc. 5 1 Weekly hours of 30 men The stem shows the 4 5 6 tens. . . and the 4 1 1 2 leaves the units 3 5 5 5 6 6 e. g. 46 is 4 tens and 6 3 0 0 1 1 2 2 3 3 3 4 4 4 units 2 8 We must show a key. 2 1 Key: 3 5 means 35 hours

Stem and Leaf Diagrams Weekly hours of 30 men 5 1 4 5 6 4 1 1 2 3 5 5 5 6 6 3 0 0 1 1 2 2 3 3 3 4 4 4 2 8 Key: 3 5 means 35 hours 2 1 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

Stem and Leaf Diagrams Finding the median and quartiles Finding these is easy because the data are in order. 5 1 Weekly hours of 30 men 4 5 6 4 1 1 2 3 5 5 5 6 6 3 0 0 1 1 2 2 3 3 3 4 4 4 2 8 Key: 3 5 means 35 hours 2 1 Median: The median is the middle item, . . . th item, the so with 30 observations we need the Tip: To 16. find the middle use where n average of 15 and is the number of items of data.

Stem and Leaf Diagrams Finding the median and quartiles Finding these is easy because the data are in order. 5 1 Weekly hours of 30 men 4 5 6 4 1 1 2 15 th 3 5 5 5 6 6 3 0 0 1 1 2 2 3 3 3 4 4 4 2 8 Key: 3 5 means 35 hours 2 1 Median: The median is the middle item, . . . th item, the so with 30 observations we need the average of 15 and 16.

Stem and Leaf Diagrams Finding the median and quartiles Finding these is easy because the data are in order. 5 1 Weekly hours of 30 men 4 5 6 4 1 1 2 3 5 5 5 6 6 15 th 16 th 3 0 0 1 1 2 2 3 3 3 4 4 4 2 8 Key: 3 5 means 35 hours 2 1 Median: The median is the middle item, . . . th item, the so with 30 observations we need the average of 15 and 16.

Stem and Leaf Diagrams Finding the median and quartiles Finding these is easy because the data are in order. 5 1 Weekly hours of 30 men 4 5 6 4 1 1 2 3 5 5 5 6 6 15 th 16 th 3 0 0 1 1 2 2 3 3 3 4 4 4 2 8 Key: 3 5 means 35 hours 2 1 Median: The median is the middle item, . . . th item, the so with 30 observations we need the average of 15 and 16. Since the 15 th and 16 th items are both 34, the median is 34. ( If the values are not the same we average them. )

Stem and Leaf Diagrams Finding the median and quartiles Finding these is easy because the data are in order. Weekly hours of 30 men 7 th 4 5 6 5 1 4 1 1 2 3 5 5 5 6 6 3 0 0 1 1 2 2 3 3 3 4 4 4 2 8 2 1 Key: 3 5 means 35 hours For the lower quartile (LQ) we first need

Stem and Leaf Diagrams Finding the median and quartiles Finding these is easy because the data are in order. Weekly hours of 30 men 7 th 4 5 6 5 1 4 1 1 2 8 th 3 5 5 5 6 6 3 0 0 1 1 2 2 3 3 3 4 4 4 2 8 2 1 Key: 3 5 means 35 hours For the lower quartile (LQ) we first need

Stem and Leaf Diagrams Finding the median and quartiles Finding these is easy because the data are in order. Weekly hours of 30 men 7 th 4 5 6 5 1 8 th 4 1 1 2 3 5 5 5 6 6 3 0 0 1 1 2 2 3 3 3 4 4 4 2 8 2 1 Key: 3 5 means 35 hours For the lower quartile (LQ) we first need The lower quartile is 32.

Stem and Leaf Diagrams Finding the median and quartiles If the values of the 7 th and 8 th observation are not the same, we interpolate to find the LQ. e. g. If we had 7 th value: 8 th value: 32 36 and we want the 7· 75 th value, we need to add 0· 75 of the gap between the 7 th and 8 th to the 7 th value. So, The gap is 36 – 32 = 4. 0. 75 of 4 is 3. LQ = 32 + 3 = 35

Stem and Leaf Diagrams Finding the median and quartiles 5 1 Weekly hours of 30 men 4 5 6 4 1 1 2 3 5 5 5 6 6 3 0 0 1 1 2 2 3 3 3 4 4 4 2 8 2 1 Key: 3 5 means 35 hours For the upper quartile (UQ), we first need ( or from the top end )

Stem and Leaf Diagrams Finding the median and quartiles 5 1 Weekly hours of 30 men 4 5 6 23 rd 4 1 1 2 3 5 5 5 6 6 3 0 0 1 1 2 2 3 3 3 4 4 4 2 8 2 1 Key: 3 5 means 35 hours For the upper quartile (UQ), we first need ( or from the top end )

Stem and Leaf Diagrams Finding the median and quartiles 5 1 Weekly hours of 30 men 4 5 6 4 1 1 2 23 rd 24 th 3 5 5 5 6 6 3 0 0 1 1 2 2 3 3 3 4 4 4 2 8 2 1 Key: 3 5 means 35 hours For the upper quartile (UQ), we first need ( or from the top end )

Stem and Leaf Diagrams Finding the median and quartiles 5 1 Weekly hours of 30 men 4 5 6 4 1 1 2 23 rd 24 th 3 5 5 5 6 6 3 0 0 1 1 2 2 3 3 3 4 4 4 2 8 2 1 Key: 3 5 means 35 hours For the upper quartile (UQ), we first need ( or from the top end ) The upper quartile is 36. The interquartile range (IQR) = UQ - LQ = 36 – 32 = 4

SUMMARY Stem and Leaf Diagrams Ø Stem and Leaf diagrams are used for small, raw data sets (not grouped data). e. g. 5 1 Weekly hours of 30 men 4 5 6 4 1 1 2 leaves 3 5 5 5 6 6 3 0 0 1 1 2 2 3 3 3 4 4 4 2 8 2 1 stem Key: 3 5 means 35 hours Ø The diagram must have a title and a key. Ø The leaves are in numerical order ( away from the stem ) continued

Stem and Leaf Diagrams Ø The median is the th piece of data. ( If necessary, average 2 data items ) Ø The quartiles are found at the and the th th position. Ø The interquartile range (IQR) = UQ - LQ ( upper quartile – lower quartile ) Ø If necessary, we can interpolate to find the LQ and UQ.

Back-to-Back Stem and Leaf Diagrams Back-to-back stem and leaf diagrams can be used to compare 2 sets of data relating to the same subject. In our example we could add the data for 30 women. Weekly hours Women 5 1 Men 4 3 4 5 6 0 4 1 1 2 8 8 6 6 5 5 3 5 5 5 6 6 4 4 4 3 3 3 2 2 2 3 0 0 1 1 2 2 3 3 3 4 4 4 8 2 8 3 3 1 0 2 1 5 5 1 4 4 2 1 Key: 5 3 means 35 hrs Notice how easily we can compare the variability of the 2 data sets Key: 3 5 means 35 hrs

Back-to-Back Stem and Leaf Diagrams Exercise January Max Temperatures 1985 to 2005 Braemar Sheffield 9 0 2 8 1 2 4 5 9 7 1 2 2 4 8 8 5 2 1 6 4 5 9 8 6 1 5 0 2 3 4 5 9 8 4 3 0 0 4 9 7 5 1 3 4 7 5 2 6 1 1 Key: 3 4 means 4· 3 C Key: 3 4 means 3· 4 C Find the medians and quartiles. What can you say about the temperatures of the 2 places?

Back-to-Back Stem and Leaf Diagrams Answer: January Max Temperatures 1985 to 2005 Braemar Sheffield 2 8 5 8 9 84 3 9 7 Key: 3 4 means 4· 3 C Median LQ UQ Braemar 4· 4 3· 6 5· 9 5 Sheffield 7· 1 5· 35 8· 1 2 6 0 5 1 1 0 1 5 6 1 9 8 7 6 5 4 3 2 1 0 1 1 4 0 2 2 5 2 4 5 9 2 4 8 9 3 4 5 4 7 Key: 3 4 means 3· 4 C The places have similar variability in temperature but Sheffield is about 2 · 7 C warmer than Braemar.

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

Stem and Leaf Diagrams SUMMARY Ø This is used for small, raw data sets (not grouped data). e. g. stem 5 1 Weekly hours of 30 men 4 5 6 leaves 4 1 1 2 3 5 5 5 6 6 3 0 0 1 1 2 2 3 3 3 4 4 4 2 8 2 1 Key: 3 5 means 35 hours Ø The diagram must have a title and a key. Ø The leaves are in numerical order ( away from the stem )

Stem and Leaf Diagrams Ø The median is the th piece of data. ( If necessary, average 2 data items ) Ø The quartiles are found at the and the th th position. Ø The interquartile range (IQR) = UQ - LQ ( upper quartile – lower quartile ) Ø If necessary, we can interpolate to find the LQ and UQ.

Stem and Leaf Diagrams Back-to-back stem and leaf diagrams can be used to compare 2 sets of data relating to the same subject. In our example we could add the data for 30 women. Women Weekly hours 5 1 Men 4 3 4 5 6 0 4 1 1 2 8 8 6 6 5 5 3 5 5 5 6 6 4 4 4 3 3 3 2 2 2 3 0 0 1 1 2 2 3 3 3 4 4 4 8 2 8 3 3 1 0 2 1 5 5 1 4 4 2 1 Key: 5 3 means 35 hours Notice how easily we can compare the variability of the 2 data sets Key: 3 5 means 35 hours