3 ShapeWe can use a graphs to look at the shape of the quantitative variable distribution.An example of a bell-shaped or normal distribution which appear often in nature:SymmetricMean, median, mode roughly equal
4 Shape: Skewed Distributions Scores from an easy exam, skewed left.Scores from a hard exam, skewed right.Non-symmetricMean < Median < ModeNon-symmetricMean > Median > Mode
7 OutliersAn outlier is an extreme value of the data (extremely high or extremely low). It is an observation value that is significantly different from the rest of the data. There may be more than one outlier in a set of data.
8 Outliers Possible Reasons for Outliers: 1. An error was made while taking the measurement or entering it into the computer.2. The individual belongs to a different group than the bulk of individuals measured.3. The outlier is a legitimate, though extreme data value.
9 Identifying Outliers (1.5*IQR) We can identify an outlier if it isLess Q1 – 1.5×IQR orGreater than Q ×IQR
10 ExampleMake a box and whisker plot of the data and identify any outliers.10, 12, 11, 15, 11, 14, 13, 17, 12, 22, 14, 11
11 The prices of a gallon of gasoline (in dollars) for selected countries in 2003 are listed below. Australia: $2.20Canada: $2.02Germany: $4.58Mexico: $2.09United States: $1.59Japan: $3.47Taiwan: $2.16Make a box and whisker plot for the gasoline prices.Which countries, if any, had gasoline prices that can be considered outliers?
12 Center Measures of center: mean, median, mode Average of the data set The middle value of a data set arranged from smallest to largestThe data value that occurs the most often, is a common measure of center for categorical data
14 CenterWhen describing data, you must decide which number is the most appropriate description of the center.Mean Median applet:Use the mean on symmetric data and the median on skewed data or data with outliers
16 Mean Absolute Deviation Variance and Standard Deviation SpreadRangeInterquartile RangeMean Absolute DeviationVariance and Standard DeviationMax value subtract minimum value(spread of all data)Interquartile range (IQR) : shows middle 50% of dataIQR = Q3 – Q1Not affected as much by outliersUse when measure of center is medianAverage distance between each data value and the meanUse when measure of center is meana measure of the “average” deviation of all observations from the mean.
17 5 number summaryComplete a 5 number summary and box and whisker plot for the following data.Number of hours spent on internet per week:12, 4, 16, 18, 1, 6, 10, 8
18 Mean Absolute Value Deviation To calculate Mean Absolute Value Deviation:Calculate the mean for the data set.Find the distance between each data value and the mean. That is, find the absolute value of the difference between each data value and the mean.Find the average of those distances.
19 Example Find the mean absolute value of the following data set: 52, 48, 60, 55, 59, 54, 58, 62
20 Standard DeviationA measure of spread is the Standard Deviation: a measure of the “average” deviation of all observations from the mean.The symbol for Standard Deviation is σ (the Greek letter sigma).
21 Standard Deviation To calculate Standard Deviation: Calculate the mean for the data set.Determine each observation’s deviation: subtract the mean from each data point. (𝑥 − 𝑥 ).Square each deviation.“Average” the squared-deviations by totaling the squared- deviation and dividing the total squared deviation by (n-1).This quantity is the Variance.Square root the result to determine the Standard Deviation.
22 Example Calculate the standard deviation of the following test scores: 15, 20, 21, 20, 36, 15, 25, 15
23 The most appropriate measure of variability depends on … The shape of the data’s distribution!If data are symmetric, with no serious outliers, use range and standard deviation.If data are skewed, and/or have serious outliers, use IQR.
24 Comparing Data Quantitative Data: through graphs Categorical Data: through two way frequency tables
25 GraphsMultiple bar graphs Multiple box and whisker plots
26 Two Way Frequency Tables These tables examine the relationships between the two categorical variables.A two-way frequency table will deal with two variables