Statistics and Modelling 2011

Topic 1: Introduction to statistical analysis Purpose – To revise and advance our understanding of descriptive statistics and how to use them.

LESSON 1 – Central tendency and Variability Activity (as students arrive): Measure each other’s heights. –Students then calculate the mean height. NOTES: What are statistics? Look at measures of Central Tendency. Measures of Variability – what do standard deviation and variance actually tell us? HW: Worksheet + Old Sigma (2 nd edition) – Pg. 148: Ex. 10.1

Statistics Statistics are ___________________________________________ ____________________. Population Mean  =Sum of all scores Population size  = N x 

Statistics Statistics are numerical values that describe the characteristics of a set of numbers (data-set). 2 types of statistics: Measures of ______ ________: Mean and median & mode. Measures of ______: Standard deviation, variance, inter-quartile range Population Mean  =Sum of all scores Population size  = N x 

Statistics Statistics are numerical values that describe the characteristics of a set of numbers (data-set). 2 types of statistics: Measures of Central Tendency: Mean and median & mode. Measures of ______: Standard deviation, variance, inter-quartile range Population Mean  =Sum of all scores Population size  = N x 

Statistics Statistics are numerical values that describe the characteristics of a set of numbers (data-set). 2 types of statistics: Measures of Central Tendency: Mean and median & mode. Measures of Variability: Standard deviation, variance, inter-quartile range Measures of Central Tendency:  Mean – easy to calculate but _________ ___ ________ _______. Population Mean  =Sum of all scores Population size  = N x 

Statistics Statistics are numerical values that describe the characteristics of a set of numbers (data-set). 2 types of statistics: Measures of Central Tendency: Mean and median & mode. Measures of Variability: Standard deviation, variance, inter-quartile range Measures of Central Tendency:  Mean – easy to calculate but sensitive to extreme values. Population Mean  =Sum of all scores Population size  = N x 

Statistics Statistics are numerical values that describe the characteristics of a set of numbers (data-set). 2 types of statistics: Measures of Central Tendency: Mean and median & mode. Measures of Variability: Standard deviation, variance, inter-quartile range Measures of Central Tendency:  Mean – easy to calculate but sensitive to extreme values. Population Mean  =Sum of all scores Population size  = N x 

2 types of statistics: Measures of Central Tendency: Mean and median & mode. Measures of Variability: Standard deviation, variance, inter-quartile range. Measures of Central Tendency:  Mean – easy to calculate but sensitive to extreme values.  Median – rank the data and find the middle entry. Advantage: More _____ than the mean because it doesn’t get dragged up or down by extreme values. Population Mean  =of all scores Population size Sum

2 types of statistics: Measures of Central Tendency: Mean and median & mode. Measures of Variability: Standard deviation, variance, inter-quartile range. Measures of Central Tendency:  Mean – easy to calculate but sensitive to extreme values.  Median – rank the data and find the middle entry. Advantage: More robust than the mean because it doesn’t get dragged up or down by extreme values. Disadvantages: Population Mean  =of all scores Population size Sum

 Mean – easy to calculate but sensitive to extreme values.  Median – rank the data and find the middle entry. Advantage: More robust than the mean because it doesn’t get dragged up or down by extreme values. Disadvantages: 1. _______________________________________________. _____________________________. 2. _____________________________. Population Mean  =of all scores Population size  N  Sum = x

 Mean – easy to calculate but sensitive to extreme values.  Median – rank the data and find the middle entry. Advantage: More robust than the mean because it doesn’t get dragged up or down by extreme values. Disadvantages: 1. Takes longer to calculate because you must rank the numbers first. 2. _____________________________. Population Mean  =of all scores Population size  N  Sum = x

 Mean – easy to calculate but sensitive to extreme values.  Median – rank the data and find the middle entry. Advantage: More robust than the mean because it doesn’t get dragged up or down by extreme values. Disadvantages: 1. Takes longer to calculate because you must rank the numbers first. 2. Not a function of all of the numbers.  The Mode - The most frequently occurring value. This is usually somewhere near the middle, but not always. Thus the mode is not always a good measure of central tendency. Population Mean  =of all scores Population size  N  Sum = x

Measures of Variability Standard deviation, variance, inter-quartile range. The standard deviation is a measure of the _____________________ _____.

Measures of Variability Standard deviation, variance, inter-quartile range. The standard deviation is a measure of the average distance from the mean. The _______ is also a measure of variability. It is ________________ _______. ___________________________________________ ____________________.

Measures of Variability Standard deviation, variance, inter-quartile range. The standard deviation is a measure of the average distance from the mean. The variance is also a measure of variability. It is ________________ _______. ___________________________________________ ____________________.

Measures of Variability Standard deviation, variance, inter-quartile range. The standard deviation is a measure of the average distance from the mean. The variance is also a measure of variability. It is the standard deviation squared. ___________________________________________ ____________________.

Measures of Variability Standard deviation, variance, inter-quartile range. The standard deviation is a measure of the average distance from the mean. The variance is also a measure of variability. It is the standard deviation squared. We actually work out the variance first, then square root it to get the standard deviation. Variance Var(X) Var(X) = The Mean of the Squared Deviations from the Mean (i.e. the average squared distance from the mean) Var(X)=

Measures of Variability Standard deviation, variance, inter-quartile range. The standard deviation is a measure of the average distance from the mean. The variance is also a measure of variability. It is the standard deviation squared. We actually work out the variance first, then square root it to get the standard deviation. Variance Var(X) Var(X) = The Mean of the Squared Deviations from the Mean (i.e. the average squared distance from the mean) Var(X)=Sum of Squared Deviations from the Sample Mean Sample size

Measures of Variability Standard deviation, variance, inter-quartile range. The standard deviation is a measure of the average distance from the mean. The variance is also a measure of variability. It is the standard deviation squared. We actually work out the variance first, then square root it to get the standard deviation. Variance Var(X) Var(X) = The Mean of the Squared Deviations from the Mean (i.e. the average squared distance from the mean) Var(X)=Sum of Squared Deviations from the Sample Mean Sample size Formula: Var(X)=

Variance Var(X) Var(X) = The Mean of the Squared Deviations from the Mean (i.e. the average squared distance from the mean) Standard Deviation s or  Standard Deviation Var(X)=Sum of Squared Deviations from the Sample Mean Sample size Formula: Var(X)=

Variance Var(X) Var(X) = The Mean of the Squared Deviations from the Mean (i.e. the average squared distance from the mean) Standard Deviation s or  Standard Deviation Formula: A measure of the average distance from the mean. Var(X)=Sum of Squared Deviations from the Sample Mean Sample size Formula: Var(X)=  Now look at the class height data, and calculate the variance & SD. HW: Do Mean & SD worksheet + Sigma (old) Ex

LESSON 2– Standard Deviation applications: Points of today:  Develop an understanding of what variance and standard deviation tell us. STARTER: Mark HW worksheet. 1.Alternative formula for variance & standard deviation. 2.Sigma (old – 2 nd edition): Pg. 151 – Ex Types of data – discrete & continuous. 4.Sigma (old – 2 nd edition): Pg. 157 – Ex (complete for HW).

Variance & standard deviation The formula: Var(X)= can be re-arranged to get: Var(X) = So Standard Deviation can be given by: s = This version of the formula is quicker to use because you don’t have to calculate the distance from the mean for each individual value. Caution: When calculating the SD from a freq. table, remember to multiply each x value by the number of times it occurs (its frequency)! Copy, then do Sigma (old) - pg. 151: Ex

Variance formula proof: Link between the 2 variance formulas: Prove that = Left hand side = = = = = = = = Right hand side

LESSON 3– Grouped data & frequency: Points of today:  Interpret displays of grouped data based on relative frequency. 1.Difference between discrete and continuous data. 2.How to calculate (estimate) the mean and standard deviation of grouped data displayed on a frequency table. Sigma (old – 2 nd edition): Ex. 10.4

Discrete and Continous data DiscreteContinuous

Discrete and Continous data DiscreteContinuous Where does its value come from?

Discrete and Continous data DiscreteContinuous Where does its value come from? Counting

Discrete and Continous data DiscreteContinuous Where does its value come from? Counting What values can it take?

Discrete and Continous data DiscreteContinuous Where does its value come from? Counting What values can it take? Whole numbers or rounded values

Discrete and Continous data DiscreteContinuous Where does its value come from? Counting What values can it take? Whole numbers or rounded values What question is being asked?

Discrete and Continous data DiscreteContinuous Where does its value come from? Counting What values can it take? Whole numbers or rounded values What question is being asked? ‘How many ?’

Discrete and Continous data DiscreteContinuous Where does its value come from? Counting What values can it take? Whole numbers or rounded values What question is being asked? ‘How many ?’ Examples:

Discrete and Continous data DiscreteContinuous Where does its value come from? Counting What values can it take? Whole numbers or rounded values What question is being asked? ‘How many ?’ Examples: Number of students who gain Excellence in the first test. Money – since we count it in dollars and cents.

Discrete and Continous data DiscreteContinuous Where does its value come from? Counting What values can it take? Whole numbers or rounded values What question is being asked? ‘How many ?’ Examples: Number of students who gain Excellence in the first test. Money – since we count it in dollars and cents.

Discrete and Continous data DiscreteContinuous Where does its value come from? CountingMeasurement What values can it take? Whole numbers or rounded values What question is being asked? ‘How many ?’ Examples: Number of students who gain Excellence in the first test. Money – since we count it in dollars and cents.

Discrete and Continous data DiscreteContinuous Where does its value come from? CountingMeasurement What values can it take? Whole numbers or rounded values All real numbers (anywhere along the number line – infinite precision) What question is being asked? ‘How many ?’ Examples: Number of students who gain Excellence in the first test. Money – since we count it in dollars and cents.

Discrete and Continous data DiscreteContinuous Where does its value come from? CountingMeasurement What values can it take? Whole numbers or rounded values All real numbers (anywhere along the number line – infinite precision) What question is being asked? ‘How many ?’‘How long ?’, ‘ How heavy ?’ Examples: Number of students who gain Excellence in the first test. Money – since we count it in dollars and cents.

Discrete and Continous data DiscreteContinuous Where does its value come from? CountingMeasurement What values can it take? Whole numbers or rounded values All real numbers (anywhere along the number line – infinite precision) What question is being asked? ‘How many ?’‘How long ?’, ‘ How heavy ?’ Examples: Number of students who gain Excellence in the first test. Money – since we count it in dollars and cents. Height Distance Weight Volume Time Copy then do Sigma (old) - pg. 157: Ex Q1 and 2.

Discrete and Continous data DiscreteContinuous Where does its value come from? What values can it take? What question is being asked? Examples:

Calculating the mean & standard deviation from a frequency table

Calculating the mean & standard deviation when data is grouped into intervals

Mean & std. deviation on your calc. On a Graphics Calculator (GC): Type: MENU, STAT, CALC (F2), SET (F6): 1VarXList:List 1 (F1) 1VarFreq:1 (F1) EXIT (goes back) Now enter the data into List 1. Press 1Var  : mean  x : sum of all values  x 2 : sum of the squares of all values x  n : standard deviation. On a Scientific Calculator: MODE 2 (puts it into STAT mode) Now enter the first data-value and press M+ (means add to memory) Repeat, pressing M+ after each: E.g.: 4 M+ 7 M+ 8 M+ etc. Every time you’ve entered in a new value it will tell you the number of values now saved. (e.g. n=3). Once all are entered, type “=“, then: SHIFT 2: 1: for the mean; 2: ( x  n ) for the SD. SHIFT Mode: Clears the memory