General Maths Unit One Univariate Data

Chapter One – Univariate Data Assigned textbook questions to be up to date by 2 nd Feb Exercise 1A Exercise 1B Exercise 1D Holiday Homework Sheet Assigned textbook questions to be up to date by 2 nd Feb Exercise 1A Exercise 1B Exercise 1D Holiday Homework Sheet

1A – Categorical Data Types of Data Data can be divided into two major groups - Categorical Data (Qualitative) Numerical Data (Quantitative) Types of Data Data can be divided into two major groups - Categorical Data (Qualitative) Numerical Data (Quantitative)

Categorical Data can be placed into one of 2 categories – 1A – Types of Data

Numerical Data is in the form of numbers and can be either – 1A – Types of Data

What type of data is……??? The number of students who walk to school. The types of vehicles that each of your parents drive. The sizes of pizza available at a pizza shop. The varying temperature outside throughout the day. What type of data is……??? The number of students who walk to school. The types of vehicles that each of your parents drive. The sizes of pizza available at a pizza shop. The varying temperature outside throughout the day. 1A – Types of Data Numerical – Discrete Categorical – Nominal Categorical – Ordinal Numerical - Continuous

Once Categorical Data has been collected, it is important to be able to summarise and display the data using – Frequency Tables Graphs – Bar Graphs or Dot Plots Once Categorical Data has been collected, it is important to be able to summarise and display the data using – Frequency Tables Graphs – Bar Graphs or Dot Plots 1A – Working with Categorical Data

Once Categorical Data has been collected, it is important to be able to calculate – Frequency Relative Frequency % Frequency Once Categorical Data has been collected, it is important to be able to calculate – Frequency Relative Frequency % Frequency 1A – Working with Categorical Data The number of times that a particular thing has occurred The relative frequency × 100

Class Hair Colour Survey Gather Data of the students in the classroom and use it to: 1. Summarise data using a frequency distribution table 2. Represent data using a bar chart 3. Find the frequency of those with brown hair 4. Find the relative frequency of those with brown hair 5. Find the % frequency of those with brown hair Class Hair Colour Survey Gather Data of the students in the classroom and use it to: 1. Summarise data using a frequency distribution table 2. Represent data using a bar chart 3. Find the frequency of those with brown hair 4. Find the relative frequency of those with brown hair 5. Find the % frequency of those with brown hair 1A – Working with Categorical Data

Hair ColourTallyTotal Brown Blonde Black Red Other 1. Summarise data using a frequency distribution table 2. Represent data using a bar chart 3. Find the frequency of those with brown hair 4. Find the relative frequency of those with brown hair 5. Find the % frequency of those with brown hair

2. Represent data using a bar chart 1. Summarise data using a frequency distribution table 2. Represent data using a bar chart 3. Find the frequency of those with brown hair 4. Find the relative frequency of those with brown hair 5. Find the % frequency of those with brown hair Remember – In a bar chart the bars don’t touch. Leave gaps!

1. Summarise data using a frequency distribution table 2. Represent data using a bar chart 3. Find the frequency of those with brown hair 4. Find the relative frequency of those with brown hair 5. Find the % frequency of those with brown hair

Eye Colour Survey Gather Data of the students in the classroom and use it to: 1. Summarise data using a frequency distribution table 2. Represent data using a bar chart 3. Find the frequency of those with brown hair 4. Find the relative frequency of those with brown hair 5. Find the % frequency of those with brown hair Eye Colour Survey Gather Data of the students in the classroom and use it to: 1. Summarise data using a frequency distribution table 2. Represent data using a bar chart 3. Find the frequency of those with brown hair 4. Find the relative frequency of those with brown hair 5. Find the % frequency of those with brown hair 1A – Working with Categorical Data Your Turn

Eye ColourTallyTotal 1. Summarise data using a frequency distribution table 2. Represent data using a bar chart 3. Find the frequency of those with brown hair 4. Find the relative frequency of those with brown hair 5. Find the % frequency of those with brown hair Your Turn Now complete the rest in your workbooks

Now do Questions from your work record Chapter 1A Q1a,b Q2b,c Q3a,b,c Q6 Q8 Q9 Q10

1B – Working with Numerical Data The remainder of this topic is concerned with Numerical Data. With Numerical Data, each data point is known as a score. Grouping Data Numerical Data can be presented as either Ungrouped Data or Grouped Data.

1B – Working with Numerical Data Grouping Data When we have a large amount of data, it’s useful to group the scores into groups or classes. When making the decision to group raw data on a frequency distribution table, choice of class (group) size matters. As a general rule, try to choose a class size so that 5 – 10 groups are formed. Find the lowest and the highest scores to decide what numbers need to be included in the groups.

1B – Working with Numerical Data Grouping Data eg1. Group the following data appropriately Tally Frequency We use an open ‘ – ‘ to include all values up to the number in the next column

1B – Working with Numerical Data Grouping Data eg2. Group the following data appropriately Tally Frequency

1B – Working with Numerical Data Histograms Similar to a bar chart with a few very important changes: Columns are drawn right against each other A gap is left at the very start of the chart If coloured in, use the same colour for all columns A polygon may be drawn to link the columns

1B – Working with Numerical Data Histograms Ungrouped Data – Data Labels appear directly under the centre of each column

1B – Working with Numerical Data Histograms Grouped Data – End points of each class appear under the edge of each column

1B – Working with Numerical Data Data Distribution We can name data according to how it’s distributed. Is it all crammed together or is there more data in certain areas?? We associate certain names with different shapes of distribution Normal – Most common score in the centre of the data Skewed – Most common score is toward one end of the data Bimodal – More than one score that is most frequent Spread – Data is spread over a wide range Clustered – Most of the data is confined to a small range

Normally Distributed Data The most common score in the centre of the data. The graph is symmetrical. 1B – Working with Numerical Data Data Distribution

Skewed Data The most common score is toward one end of the data. Most data toward the left – Postively Skewed Most data toward the right – Negatively Skewed 1B – Working with Numerical Data Data Distribution

Bimodal Data More than one score that is most frequent This looks like two peaks on the graph 1B – Working with Numerical Data Data Distribution

Spread Data Data is rather evenly spread over a wide range 1B – Working with Numerical Data Data Distribution

Clustered Data Most of the data is confined to a small range 1B – Working with Numerical Data Data Distribution

Now do Questions from your work record Chapter 1B Q1 Q2 Q4 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13

1D – Measures of Centre Would you agree that one of the main things statisticians do with a set of data, is to find the average, the middle or the most commonly occurring score? We call these values: The Mean – The Average of all scores. The Median – The middle score in a set of ordered data. The Mode – The score which occurs most often. We can find these values as follows…..

The Mean

The Median

The Mode The score which occurs most frequently There can be one or more than one score which occurs most frequently, in these cases they are both modes – list them both. eg. Find the mode of the data set: You may wish to write the scores in order to ensure all data is accounted for but this is not necessary Mode = 7

eg. Find the Mean, Median and Mode of the following set of data

Now do Questions from your work record Chapter 1D - Q1b,c Q2 Q3 Q4 Q8 Q9a-f Q10 Q11a-d Q14 Q15 Q16b

Holiday Homework Reminder! Assigned textbook questions to be up to date by 2 nd Feb Exercise 1A Exercise 1B Exercise 1D Holiday Homework Sheet Assigned textbook questions to be up to date by 2 nd Feb Exercise 1A Exercise 1B Exercise 1D Holiday Homework Sheet

Review Question I surveyed 15 students and asked them the score they got on their test (out of 60). The following data was obtained. 35, 45, 41, 46, 38, 56, 59, 43, 46, 45, 51, 53, 43, 46, 50 What type of data is this called? Find the Mean, Mode and Median. Group this data in an appropriate class size and represent this using a frequency table. Draw a histogram of the data. What do we call the distribution of this data? I surveyed 15 students and asked them the score they got on their test (out of 60). The following data was obtained. 35, 45, 41, 46, 38, 56, 59, 43, 46, 45, 51, 53, 43, 46, 50 What type of data is this called? Find the Mean, Mode and Median. Group this data in an appropriate class size and represent this using a frequency table. Draw a histogram of the data. What do we call the distribution of this data?

I surveyed 15 students and asked them the current age of their mothers in whole years. The following data was obtained. 35, 45, 41, 46, 38, 56, 59, 43, 46, 45, 51, 53, 43, 46, 50 Group this data in an appropriate class size and represent this using a frequency table. Smallest number = 35, Largest number = 59 Draw a histogram of the data. What do we call the distribution of this data? Normally Distributed I surveyed 15 students and asked them the current age of their mothers in whole years. The following data was obtained. 35, 45, 41, 46, 38, 56, 59, 43, 46, 45, 51, 53, 43, 46, 50 Group this data in an appropriate class size and represent this using a frequency table. Smallest number = 35, Largest number = 59 Draw a histogram of the data. What do we call the distribution of this data? Normally Distributed

Now Do Review Worksheet Then problems from text 1A – 1e, 3d, 13 1B – 13 1D – 5, 6, 7

1E – Measures of Variability Data can be viewed in ways other than just finding the middle of the data (median/mode/mean). To give us a truer representation of a set of data, we can calculate how spread out our data is using: The Range The Interquartile Range (IQR) The Standard Deviation The Variance Data can be viewed in ways other than just finding the middle of the data (median/mode/mean). To give us a truer representation of a set of data, we can calculate how spread out our data is using: The Range The Interquartile Range (IQR) The Standard Deviation The Variance

1E – Measures of Variability Range 1. Find the largest value in the set of data X max 2. Find the smallest value in the set of data X min 3. Subtract the smallest value from the largest value Range = Xmax – Xmin eg What would the range be? 1. Find the largest value in the set of data X max 2. Find the smallest value in the set of data X min 3. Subtract the smallest value from the largest value Range = Xmax – Xmin eg What would the range be? 8 – 2 = 6

1E – Measures of Variability Interquartile Range (IQR) To overcome problems due to extreme values, we can exclude the top & bottom quarters of the data to find the range of the remaining data. The lower quartile (Q1) is the number occurring ¼ of the way through the data – the 25 th percentile. The upper quartile (Q3) is the number occurring ¾ of the way through the data – the 75 th percentile. The IQR is the difference between these values, so can be found using: IQR = Q3 – Q1 To overcome problems due to extreme values, we can exclude the top & bottom quarters of the data to find the range of the remaining data. The lower quartile (Q1) is the number occurring ¼ of the way through the data – the 25 th percentile. The upper quartile (Q3) is the number occurring ¾ of the way through the data – the 75 th percentile. The IQR is the difference between these values, so can be found using: IQR = Q3 – Q1

1E – Measures of Variability Interquartile Range (IQR) Steps to calculate the IQR 1.Arrange the data in order of size 2.Divide the data in two by finding the median 3.Using the lower half of the data, find the lower quartile (Q1) by dividing this in two and find the midpoint 4.Repeat this for the upper half of data to find Q3 5.Calculate the IQR by finding Q3 – Q1. Eg. Calculate the IQR of the data: Steps to calculate the IQR 1.Arrange the data in order of size 2.Divide the data in two by finding the median 3.Using the lower half of the data, find the lower quartile (Q1) by dividing this in two and find the midpoint 4.Repeat this for the upper half of data to find Q3 5.Calculate the IQR by finding Q3 – Q1. Eg. Calculate the IQR of the data:

1E – Measures of Variability Interquartile Range (IQR) Our Data: Arrange in size order: Divide in half to find Median Find Q Find Q Find IQR = Q3 – Q1 = 7 – 2 = 5 Our Data: Arrange in size order: Divide in half to find Median Find Q Find Q Find IQR = Q3 – Q1 = 7 – 2 = 5

Shows how much variation there is from the average. A low standard deviation indicates that the data points tend to be very close to the mean. A high standard deviation indicates that the data points are spread out over a large range of values. Standard Deviation = Shows how much variation there is from the average. A low standard deviation indicates that the data points tend to be very close to the mean. A high standard deviation indicates that the data points are spread out over a large range of values. Standard Deviation = 1E – Measures of Variability Standard Deviation

The variance is simply the standard deviation squared. 1E – Measures of Variability Variance

Relax! We can use our calculator to find these values too – make sure you know how to use it!! Video Relax! We can use our calculator to find these values too – make sure you know how to use it!! Video 1E – Measures of Variability Information Overload?? Use these to find Range Use these to find IQR Variance = Standard Deviation (Standard Deviation) 2

We can find the measures of centre (Mean, Mode, Median) and the measures of spread (Range, IQR, Standard Deviation, Variance) of grouped data by first finding the midpoint of each group. 1E – Working with Grouped Data Data Frequency Midpoint These are the columns we enter into our calculator

eg. Find the mean and the standard deviation of the grouped data Find the midpoint of each group. Enter this new table into our calculator. Menu -> Statistics -> Enter the data into 2 lists eg. Find the mean and the standard deviation of the grouped data Find the midpoint of each group. Enter this new table into our calculator. Menu -> Statistics -> Enter the data into 2 lists 1E – Working with Grouped Data

Find the mean and the standard deviation of the grouped data Calc  One-Variable Choose XList: list1 (where your values are) And Freq: list2 (where your frequencies are) Find the mean and the standard deviation of the grouped data Calc  One-Variable Choose XList: list1 (where your values are) And Freq: list2 (where your frequencies are)

Now Do Exercise 1E 1c, 1e, 4a, 4b, 4c, 10, 12, 13, 14, 15, 16, 19

Instead of using a frequency table, we can also display our data using a stem and leaf plot. Similar to frequency tables, we can choose an appropriate group size in which to represent our data, usually using a class size of 5 or 10. Instead of using a frequency table, we can also display our data using a stem and leaf plot. Similar to frequency tables, we can choose an appropriate group size in which to represent our data, usually using a class size of 5 or 10. 1F – Stem and Leaf Plots Example: Stem & Leaf representation of the following data (class size of 10): 6, 7, 8, 10, 12, 13, 14, 17, 17, 17, 18, 19, 21, 23, 24, 24, 25, 27, 31, 31, 32, 36, 36, 39, 41, 45, 45, 46, 49, 50

Example: Stem & Leaf representation of the following data (class size of 10): 6, 7, 8, 10, 12, 13, 14, 17, 17, 17, 18, 19, 21, 23, 24, 24, 25, 27, 31, 31, 32, 36, 36, 39, 41, 45, 45, 46, 49, 50 Now lets try using a class size of 5….. Separate each class using * beside the stem number for the upper end of the group Include a key, this time with an example for both lower (no *) and upper ends (with *)

1F – Stem and Leaf Plots Your Turn: Organise the following data onto a stem and leaf plot using class size of , 12, 16, 21, 24, 27, 29, 31, 33, 34 Now try again using a class size of 5. Your Turn: Organise the following data onto a stem and leaf plot using class size of , 12, 16, 21, 24, 27, 29, 31, 33, 34 Now try again using a class size of 5.

Now Do Exercise 1F 1, 3, 4, 7, 8, 9, 10, 11, 12

1G – 5-Number Summary

eg1. Write the 5-Number summary for the following set of data eg1. Write the 5-Number summary for the following set of data So the 5-Number summary is: 3, 6, 10, 18, 20

1G – 5-Number Summary eg2. Write the 5-Number summary for the following set of data eg2. Write the 5-Number summary for the following set of data So the 5-Number summary is: 2, 4, 7.5, 10, Arrange in order:

Eg2 cont’d. Check your answer using the calculator Our Answer was 2, 4, 7.5, 10, 18 Eg2 cont’d. Check your answer using the calculator Our Answer was 2, 4, 7.5, 10, 18 Enter data into list1 Select the list your data is in as XList minX Q1 Med Q3 maxX

1G - Boxplots We can represent the 5-number summary on a boxplot. Boxplots are: Always drawn to scale Drawn with labels (Xmin, Q1 etc) or with a scaled & labelled axis running alongside the plot We can represent the 5-number summary on a boxplot. Boxplots are: Always drawn to scale Drawn with labels (Xmin, Q1 etc) or with a scaled & labelled axis running alongside the plot Scale Xmax Xmin Q1 Q3 Median

1G - Boxplots eg3. Using the 5-figure summary from example 2, sketch it’s boxplot. The 5-Number summary is: 2, 4, 7.5, 10, 18 Xmax XminQ1 Q3 Median

Eg3 cont’d. We can also use the calculator to help sketch the plot. Once the data has been input to the calculator, do the following: Eg3 cont’d. We can also use the calculator to help sketch the plot. Once the data has been input to the calculator, do the following: To select point on the plot click:

Now Do Exercise 1G 2, 4, 5, 8, 9, 10, 11, 12, 13, 14, 15

1H - Comparing sets of data Back-to-back Stem & Leaf Plots Used to compare 2 similar sets of data The two sets of data share the same central stem Data is ordered from smallest to largest around the central stem Used to compare 2 similar sets of data The two sets of data share the same central stem Data is ordered from smallest to largest around the central stem

1H - Comparing sets of data Back-to-back Stem & Leaf Plots Create a back-to-back Stem & Leaf for the two sets of data (using a class size of 10) : Sample A: 4, 6, 7, 10, 12, 15, 19, 24 Sample B: 5, 7, 9, 9, 13, 16, 20, 22 Create a back-to-back Stem & Leaf for the two sets of data (using a class size of 10) : Sample A: 4, 6, 7, 10, 12, 15, 19, 24 Sample B: 5, 7, 9, 9, 13, 16, 20, , 6, 4 9, 5, 2, 0 4 5, 7, 9, 9 3, 6 0, 2 Remember to start each line from the centre and work your way out Always include a key!

1H - Comparing sets of data Back-to-back Stem & Leaf Plots Create a back-to-back Stem & Leaf for the two sets of data (using a class size of 5) : Sample A: 4, 6, 7, 10, 12, 15, 19, 24 Sample B: 5, 7, 9, 9, 13, 16, 20, 22 Create a back-to-back Stem & Leaf for the two sets of data (using a class size of 5) : Sample A: 4, 6, 7, 10, 12, 15, 19, 24 Sample B: 5, 7, 9, 9, 13, 16, 20, , 6 2, 0 9, 5 4 5, 7, 9, , 2 Remember to start each line from the centre and work your way out * 1* Always include a key!

1H - Comparing sets of data Back-to-back Stem & Leaf Plots Find the 5-number summary for each set of data Xmin, Q1, Median, Q3, Xmax Group One 2, 10, 19, 24, 27 Group Two 1, 7, 14, 23, 27 Find the 5-number summary for each set of data Xmin, Q1, Median, Q3, Xmax Group One 2, 10, 19, 24, 27 Group Two 1, 7, 14, 23, 27

1H - Comparing sets of data Side-by-Side Box Plots Recall Boxplot – Two or more sets of data compared using side-by-side boxplots. The boxplots share a common scale so they can be compared appropriately Recall Boxplot – Two or more sets of data compared using side-by-side boxplots. The boxplots share a common scale so they can be compared appropriately Xmax Xmin Q1 Q3 Median

1H - Comparing sets of data Side-by-Side Box Plots Compare the two box plots……..what can be said about the data?

1H - Comparing sets of data Side-by-Side Box Plots eg. Two sets of data gave the following 5-figure summaries. Sample A8, 10, 15, 21, 23 Sample B5, 12, 18, 22, 25 Compare the two using side-by-side box plots. eg. Two sets of data gave the following 5-figure summaries. Sample A8, 10, 15, 21, 23 Sample B5, 12, 18, 22, 25 Compare the two using side-by-side box plots. Sample A Sample B

Now Do Exercise 1H 1 – 10; 14

Revision Problems Univariate Data

Revision Question One The following table shows the dinner bookings from a local restaurant over an evening. What is the frequency of a group having 3 people? What is the relative frequency of a group with 3 people? What is the percentage frequency of a group with 3 people? What is the total number of people who attended the restaurant that evening? Draw a histogram of the data. What is the average group size? Group Size12345 Frequency

Revision Question Two The stem and leaf plot below shows the height of a group of 20 students. State the minimum height. State the median height. State the Mode. State the IQR. State the Standard Deviation. How many people over 172cm tall? What is the relative frequency of a person who is 166cm tall? What type of distribution is this? StemLeaf 15*8, 9 160, 2, 4 16*5, 6, 6, 8, 9 171, 3, 4, 4, 4 17*5, 8, 9 181, 4 Key: 15* 8 = 158cm 16 0 = 160cm

Revision Question Three The batting scores of two batsmen were collected over a cricket season. Their results are compared on the boxplots below. Which batsman had the highest score? What was this score? Write a 5-number summary for each Batsman A & Batsman B. Which batsman had the best median performance? Which batsman had the smallest range? What scores made up the top 50% of the runs by Batsman A? What scores made up the bottom 25% of the runs of Batsman B? Which batsman had the best overall result? Explain.

Revision Question Four Consider the following data that shows the heights (in cm) of 40 girls who are competing in trials to form a basketball squad. Using your calculator - * Find the points of central tendency, that is the Mean, Mode and Median. * Find the measures of variability, that is the Range, IQR, Standard Deviation & Variance. * Find the 5-number summary and use this to draw the boxplot of the data. Draw a frequency table of the data, using a class size of 5. Represent the data on a histogram