Measures of Spread Chapter 3.3 – Tools for Analyzing Data I can: calculate and interpret measures of spread MSIP/Home Learning: p. 168 #2b, 3b, 4, 6, 7, 10

What is spread? Measures of central tendency do not always tell you everything! The histograms have identical mean and median, but the spread is different Spread tells you how widely the data are dispersed

Why worry about spread? Spread is how closely the values cluster around the middle value Less spread means you have greater confidence that values will fall within a particular range Important for making predictions

Measures of Spread There are 3 Measures of Central Tendency: Mean, Median, Mode All measure the centre of a set of data We will also study 3 Measures of Spread: Range, Interquartile Range, Standard Deviation All measure how spread out data is  Smaller value = less spread, more consistent

Measures of Spread 1) Range = (max) – (min)  Indicates the size of the interval that contains 100% of the data 2) Interquartile Range  IQR = Q3 – Q1  where Q1 is the lower half median and Q3 is the upper half median  Indicates the size of the interval that contains the middle 50% of the data

Quartiles Example Q2 = 41Median Q1 = 36Lower half median Q3 = 46Upper half median IQR = Q3 – Q1 = 46 – 36 = 10 (50% of data is 10 units apart) If a quartile occurs between 2 values, it is calculated as the average of the two values

A More Useful Measure of Spread Range is very basic  Does not take clusters or outliers into account Interquartile range is somewhat useful  Takes clusters and outliers into account  Visual in Box-and-Whisker Plot Standard deviation is very useful  The average distance from the mean for all data points

Standard Deviation 1. Find the mean (average) 2. Find the deviation for each data point (data) – (mean) 3. Square the deviations 4. Average the squares of the deviations (this is called the variance) 5. Take the square root of the variance

Deviation The mean of these numbers is 48 Deviation = (data) – (mean) The deviation for 24 is = The deviation for 84 is = 36

Example of Standard Deviation mean = ( ) / 4 = 31 σ² = (26–31)² + (28-31)² + (34-31)² + (36-31)² 4 σ² = σ² = 17 σ = √17 = 4.1

Measure of Spread - Recap Measures of Spread are numbers indicating how spread out data is Smaller value for any measure of spread means data is more consistent 1) Range = Max – Min 2) Interquartile Range: IQR = Q3 – Q1  Q1 = first half median  Q3 = second half median 3) Standard Deviation  Find mean (average)  Find all deviations (data) – (mean)  Square all and average them - this is variance or σ 2  Take the square root to get std. dev. σ

Standard Deviation σ² (lower case sigma squared) is used to represent variance σ is used to represent standard deviation σ is commonly used to measure the spread of data, with larger values of σ indicating greater spread we are using a population standard deviation

Standard Deviation with Grouped Data grouped mean = (2×2 + 3×6 + 4×6 + 5×2) / 16 = 3.5 deviations:  2: 2 – 3.5 = -1.5  3: 3 – 3.5 = -0.5  4: 4 – 3.5 = 0.5  5: 5 – 3.5 = 1.5 σ² = 2(-1.5)² + 6(-0.5)² + 6(0.5)² + 2(1.5)² 16 σ² = σ = √ = 0.9 Hours of TV 2345 Frequency2662

MSIP / Home Learning Read through the examples on pages Complete p. 168 #2b, 3b, 4, 6, 7, 10 You are responsible for knowing how to do simple examples by hand (~6 pieces of data) We will use technology (Fathom/Excel) to calculate larger examples Have a look at your calculator and see if you have this feature (Σσn and Σσn-1)

Normal Distribution 3.4 – Tools for Analyzing Data Learning goal: Determine the % of data within intervals of a Normal Distribution Due now: p. 168 #2b, 3b, 4, 6, 7, 10 MSIP / Home Learning: p. 176 #1, 3b, 6, 8-10

Histograms Histograms can be skewed... Right-skewed Left-skewed

Histograms... or symmetrical

Normal? A normal distribution is a histogram that is symmetrical and has a bell shape Used quite a bit in statistical analysis Also called a Gaussian Distribution Symmetrical with equal mean, median and mode that fall on the line of symmetry of the curve

A Real Example the heights of 600 randomly chosen Canadian students from the “Census at School” data set the data approximates a normal distribution

The % Rule Area under curve is 1 (i.e. it represents 100% of the population surveyed) Approx 68% of the data falls within 1 standard deviation of the mean Approx 95% of the data falls within 2 standard deviations of the mean Approx 99.7% of the data falls within 3 standard deviations of the mean

Distribution of Data 34% 13.5% 2.35% 68% 95% 99.7% xx + 1σx + 2σx + 3σx - 1σx - 2σx - 3σ 0.15%

Normal Distribution Notation The notation above is used to describe the Normal distribution where x is the mean and σ² is the variance (square of the standard deviation) e.g. X~N (70,8 2 ) describes a Normal distribution with mean 70 and standard deviation 8 (our class at midterm?)

An example Suppose the time before burnout for an LED averages 120 months with a standard deviation of 10 months and is approximately Normally distributed. What is the length of time a user might expect an LED to last with: a) 68% confidence? b) 95% confidence? So X~N(120,10 2 )

An example cont’d 68% of the data will be within 1 standard deviation of the mean This will mean that 68% of the bulbs will be between 120–10 = 110 months and = 130 months So 68% of the bulbs will last months 95% of the data will be within 2 standard deviations of the mean This will mean that 95% of the bulbs will be between 120 – 2×10 = 100 months and ×10 = 140 months So 95% of the bulbs will last months

Example continued… Suppose you wanted to know how long 99.7% of the bulbs will last? This is the area covering 3 standard deviations on either side of the mean This will mean that 99.7% of the bulbs will be between 120 – 3×10 months and ×10 So 99.7% of the bulbs will last months This assumes that all the bulbs are produced to the same standard

Example continued… 34% 13.5% 2.35% 95% 99.7% months

Percentage of data between two values The area under any normal curve is 1 The percent of data that lies between two values in a normal distribution is equivalent to the area under the normal curve between these values See examples 2 and 3 on page 175

Why is the Normal distribution so important? Many psychological and educational variables are distributed approximately normally:  height, reading ability, memory, IQ, etc. Normal distributions are statistically easy to work with  All kinds of statistical tests are based on it Lane (2003)

Exercises Complete p. 176 #1, 3b, 6,

References Lane, D. (2003). What's so important about the normal distribution? Retrieved October 5, 2004 from bution.html Wikipedia (2004). Online Encyclopedia. Retrieved September 1, 2004 from