© 2003 Prentice-Hall, Inc. Basic Business Statistics Chapter 3 Numerical Descriptive Measures Chapters Objectives: Learn about Measures of Center. How to calculate mean, median and midrange Learn about Measures of Spread Learn how to calculate Standard Deviation, IQR and Range Learn about 5 number summaries Learn how to create Box Plots

© 2003 Prentice-Hall, Inc. Chapter Topics Measures of Central Tendency Mean, Median, Midrange, Geometric Mean Quartile Measure of Variation Range, Interquartile Range, Variance and Standard Deviation, Coefficient of Variation Shape Symmetric, Skewed, Using Box-and-Whisker Plots

© 2003 Prentice-Hall, Inc. Chapter Topics The Empirical Rule and the Bienayme- Chebyshev Rule Coefficient of Correlation Pitfalls in Numerical Descriptive Measures and Ethical Issues (continued)

© 2003 Prentice-Hall, Inc. Summary Measures Central Tendency Mean Median Mode Quartile Geometric Mean Summary Measures Variation Variance Standard Deviation Coefficient of Variation Range

© 2003 Prentice-Hall, Inc. The Second Step of Data Analysis We looked at displaying data using graphs. But numeric summaries help us to compare variables and to talk about relationships among variables in precise (exact) ways. After drawing the graph, it is usual to calculate summary values. This Lesson considers a different numerical summaries.

© 2003 Prentice-Hall, Inc. Summaries of Center and Spread Can you think of a way of estimating the Center of this histogram or How wide it is (Spread or Variance) Measure of Center Measureo f Spread MeanStandard Deviation MediaIQR MidrangeRange The shape that appears over the histogram is called the normal distribution shape This shape is used to estimate the shape of the histogram

© 2003 Prentice-Hall, Inc. Finding the Central Value We summarize the distribution with mean median and midrange The mean median and midrange have different idea of what is the center, and behaves differently The Mode is also used however this is not a good measure of the central value Geometric Mean will not be covered in this course

© 2003 Prentice-Hall, Inc. Measures of Central Tendency Central Tendency MeanMedianMidrange Geometric Mean

© 2003 Prentice-Hall, Inc. Mean (Arithmetic Mean) Mean (Arithmetic Mean) of Data Values Sample mean Population mean Sample Size Population Size

© 2003 Prentice-Hall, Inc. Mean (Arithmetic Mean) The Most Common Measure of Central Tendency Affected by Extreme Values (Outliers) (continued) Mean = 5Mean = 6

© 2003 Prentice-Hall, Inc. Approximating the Arithmetic Mean Used when raw data are not available Mean (Arithmetic Mean) (continued)

© 2003 Prentice-Hall, Inc. How to compute the Mean? Step 1: Add all the data points. This is called the SUM or the Total Step 2: Count the number of Data points. This is sometimes called the count Step 3: Divide The Total by the Number of Data points Calculate the mean for the following data points. 5,20,15,3,7 Revision: 5.1 Know how to compute the mean of a collection of data values.

© 2003 Prentice-Hall, Inc. Median Robust Measure of Central Tendency Not Affected by Extreme Values In an Ordered Array, the Median is the ‘Middle’ Number If n or N is odd, the median is the middle number If n or N is even, the median is the average of the 2 middle numbers Median = 5

© 2003 Prentice-Hall, Inc. Example: Find median of the following numbers Step 1 Order the data Step 2 There are 7 values n is odd Step 3 a Median is the middle number Median is13.9 If we added 35.7 to the above numbers (data set). What would the median now be? Step 3 b Median is average of the two middle numbers 5 position position13.9 Median ='( )/2= Step 1 Order the data n is even Step 2 There are 8 values Calculate the median for the following data points. 5,20,15,3,7 Calculate the median for the following data points. 5,20,15,3,7,10 MEDIAN MEDIAN between 13.9 and 14.1

© 2003 Prentice-Hall, Inc. Mode A Measure of Central Tendency Value that Occurs Most Often Not Affected by Extreme Values There May Not Be a Mode There May Be Several Modes Used for Either Numerical or Categorical Data Mode = No Mode

© 2003 Prentice-Hall, Inc. Why the mode is not really a measure of center Often listed among measures of center. Is it really a central value? 1. The mode of a categorical variable is the category with the highest frequency. It is not a center in because the categories of a categorical variable have no specific order we can choose to place the modal category in a bar chart on the right, on the left, or in the middle.. 2. When a continuous distribution is unimodal and symmetric, the single, central mode is often near to other measures of center. 3. Modes of data measured on quantitative variables are rarely useful as measures of center. It is a coincidence when two quantitative measurements agree exactly, counting such occurrences tells us little about the variable. As a consequence, statistics packages rarely compute or report modes.

© 2003 Prentice-Hall, Inc. Geometric Mean Useful in the Measure of Rate of Change of a Variable Over Time Geometric Mean Rate of Return Measures the status of an investment over time

© 2003 Prentice-Hall, Inc. Example An investment of $100,000 declined to $50,000 at the end of year one and rebounded back to $100,000 at end of year two:

© 2003 Prentice-Hall, Inc. How to compute the Midrange The range of the data is defined as the difference between the Maximum and Minimum (Range = Max – Min) What would the Range be in the previous example? What would the midrange be? (max + min)/2 DISADVANTAGE: - one extreme value can make the midrange very large, so it does not really represent the data. If you deleted 45.8 and added 1000 to the previous example what would the midrange be? Calculate the midrange and range for the following data points. 5,20,15,3,7

© 2003 Prentice-Hall, Inc. Summary Measures of center Measures of center are the most commonly used (and misused) summaries of data. If you understand how they differ and what they really say you can avoid misunderstanding data or being fooled by inappropriate summaries. 5.1 Check your knowledge of centers.

© 2003 Prentice-Hall, Inc. Quartiles Split Ordered Data into 4 Quarters Position of i-th Quartile Q1 and Q3 are Measures of Noncentral Location Q2= Median, a Measure of Central Tendency The calculation will be covered in the IQR slides. 25% Do not use this formula

© 2003 Prentice-Hall, Inc. Know the differing properties of the mean, median, and midrange The most common summary value describing a distribution of data values is some measure of a typical or central value. This can be the mean, median or midrange Revision 5.1 Know the differing properties of the mean, median, and midrange of a set of data values You have calculate the 3 measures of center (mean, median and midrange) for the following data points. 5,20,15,3,7. 1.Change 15 to 19 and recalculate the mean, median and midrange. Which measures of center have changed? 2. Change 20 to 200 and recalculate the mean, median and midrange. Which measures of center have changed? The median is not effected by outliers like the midrange and Mean The mean is effected by very small changes to data

© 2003 Prentice-Hall, Inc. Why Measures Spread? Spread measures how often something varies. Imagine you go to a restaurant and received such very good meal, you return for the same meal the next day. However, this time the food tastes very bad. You never go back. The standard of quality has varied. The restaurant management needs to be able to control the quality of food. A key aspect of management is to reduce variability, and provide a consistent quality of service or product The first step to control variability is to measure it. Then you can work out what causes the variability Then work out how to reduce variability. 5.2 Compare ways to find the spread of a distribution

© 2003 Prentice-Hall, Inc. Days of Week Day 1Day 2Day 3Day 4Day 5 No Cakes baked Bakery A Bakery B Mean A Mean B Spread A is 135. Variance is very high Spread B is 23 The two bakeries have been baking too many or too few cakes each day. Why?

© 2003 Prentice-Hall, Inc. Different Summaries of Spread Summaries of Spread include Range Interquartile Range (IQR) and Standard Deviation These summaries highlight different aspects of the distribution and have different values But they all summarize how the values are spread out. We have already seen that Range = Max - Min

© 2003 Prentice-Hall, Inc. Measures of Variation Variation VarianceStandard DeviationCoefficient of Variation Population Variance Sample Variance Population Standard Deviation Sample Standard Deviation Range Interquartile Range

© 2003 Prentice-Hall, Inc. Range Measure of Variation Difference between the Largest and the Smallest Observations: Ignores How Data are Distributed Range = = Range = = 5

© 2003 Prentice-Hall, Inc. Measure of Variation Also Known as Midspread Spread in the middle 50% Difference between the First and Third Quartiles Not Affected by Extreme Values Interquartile Range Data in Ordered Array:

© 2003 Prentice-Hall, Inc. 5 Steps to calculate the Interquartile Range (IQR) Step 1Put the data in Order Step 2 Divide the dataset in two. Dataset 1 and Dataset 2. NOTE: If n is odd the median should be included in both data sets Step 3Calculate Q1 (25 th Percentile) If n of Dataset 1 is odd, the Q1 is the middle number of Dataset A If n of Dataset 1 is even, the median is the average of the 2 middle numbers Step 4Calculate Q3 (75th percentile) As per step 3 Step 5 Find The Interquartile Range (IQR) IQR= Q3-Q1

© 2003 Prentice-Hall, Inc. 5 Steps to Calculate Q1, Q3 and IQR when n is odd In order to calculate IQR we need to to first of all calculate Q1 and Q3. We are going to calculate the Q1, Q3 and IQR for the following dataset: 7,8,25,11,13,5,6 We require to know: Median (Q2 or 50 th percentile), Q1 (25th Percentile), Q3 (75 th percentile) Step 1Order Data

© 2003 Prentice-Hall, Inc. Step 2Divide the dataset in two Divide the above data set into Dataset 1 and Dataset 2. NOTE: If n is odd the median should be included in both data sets Dataset 1 is used to calculate Q1 and Dataset 2 is used to calculate Q3 Dataset 1 For Q1Data Set 2 For Q Step 3Calculate Q1 (25 th Percentile) N is even so the Q1 is the average of the 2 middle numbers which is position 2 and 3 Q1 = (6+7)/2 =6.5 (This is the same method used to calculate the median for a even number of data points) 5 Steps Calculate Q1, Q3 and IQR when n is odd (Cont)

© 2003 Prentice-Hall, Inc. Step 4 Calculate Q3 (75th percentile) N is even for dataset 2, the median is the average of the 2 middle numbers which is position 2 and 3 Thus, Q3 = (11+13)/2 =12 Step 5 Find The Interquartile Range (IQR) IQR= Q3-Q1 = =5.5 5 Steps to Calculate Q1, Q3 and IQR when n is odd(Cont) Calculate the Q1, Q3 and IQR for the following data points. 5,20,15,3,7,16,4

© 2003 Prentice-Hall, Inc. 5 Steps to Calculate Q1, Q3 and IQR when n is EVEN When n is even we we use almost the same process to calculate Q1 Q3 and IQR. In order to calculate IQR we need to to first of all calculate Q1 and Q3. We are going to calculate the Q1, Q3 and IQR for the following dataset: 7,8,25,11,13,5,6 and 30 We require to know: Q1 (25th Percentile), Q3 (75th Step 1Order Data

© 2003 Prentice-Hall, Inc. Step 2Divide the dataset in two Divide the above data set into Dataset 1 and Dataset 2. Dataset 1 is used to calculate Q1 and Dataset 2 is used to calculate Q3 Dataset 1 For Q1Dataset 2 For Q Step 3Calculate Q1 (25 th percentile) N is even so the Q1 is the average of the 2 middle numbers which is position 2 and 3 Q1 = (6+7)/2 =6.5 (This is the same method used to calculate the median for a even number of data points) 5 Steps: Calculate Q1, Q3 and IQR when n is even (Cont)

© 2003 Prentice-Hall, Inc. Step 4 Calculate Q3 (75th percentile) N is even so the Q1 is the average of the 2 middle numbers which is position 2 and 3 Thus, Q3 = (13+25)/2 =19 Step 5 Find The Interquartile Range (IQR) IQR= Q3-Q1 =19-6.5= Steps to Calculate Q1, Q3 and IQR when n is even (Cont) Calculate the Q1, Q3 and IQR for the following data points. 5,20,15,3,7,16,4,30

© 2003 Prentice-Hall, Inc. Important Measure of Variation Shows Variation about the Mean Sample Variance: Population Variance: Variance

© 2003 Prentice-Hall, Inc. Standard Deviation Most Important Measure of Variation Shows Variation about the Mean Has the Same Units as the Original Data Sample Standard Deviation: Population Standard Deviation:

© 2003 Prentice-Hall, Inc. Approximating the Standard Deviation Used when the raw data are not available and the only source of data is a frequency distribution Standard Deviation (continued)

© 2003 Prentice-Hall, Inc. Standard Deviation The sum of the deviations from the mean, divided by the count minus 1 is the variance The square root of the variance is a measure of spread called the Standard Deviation 5.2 Know how to calculate the Standard Deviation

© 2003 Prentice-Hall, Inc. How to Calculate the Standard Deviation? Steps to Calc Standard Deviation 1: Calculate the Mean =Sum/Count Steps to Calc Standard Deviation 1: Calculate the Mean =Sum/Count 2: Enter the mean on each line on the table Steps to Calc Standard Deviation 1: Calculate the Mean =Sum/Count 2: Enter the mean on each line on the table 3: Take the each data point from the mean Steps to Calc Standard Deviation 1: Calculate the Mean =Sum/Count 2: Enter the mean on each line on the table 3: Take the each data point from the mean 4: Square the answer to Step 3 (Multiply each data point by each other) Steps to Calc Standard Deviation 1: Calculate the Mean =Sum/Count 2: Enter the mean on each line on the table 3: Take the each data point from the mean 4: Square the answer to Step 3 (Multiply each data point by each other) 5: Sum the all the data points 6: Divide the total for Step 5 by (Count -1) 7: Calculate the square root of Step 6 Steps to Calc Standard Deviation 1: Calculate the Mean =Sum/Count 2: Enter the mean on each line on the table 3: Take the each data point from the mean 4: Square the answer to Step 3 (Multiply each data point by each other) 5: Sum the all the data points 6: Divide the total for Step 5 by (Count -1) Steps to Calculate Standard Deviation 1: Calculate the Mean =Sum/Count 2: Enter the mean on each line on the table 3: Take the each data point from the mean 4: Square the answer to Step 3 (Multiply each data point by each other) 5: Sum the all the data points Calculate the Standard Deviation for the following data points. 5,20,15,3,7,16,4

© 2003 Prentice-Hall, Inc. Histograms to display measures of spread When the distribution of data is unimodal and symmetric, the middle 38% of the distribution is about one standard deviation wide. This rule of thumb provides a way to display the standard deviation, but it only applies to unimodal, symmetric distributions. We can display common measures of spread in a Histogram. 5.2 Learn a way to display the spread in a Histogram. 5.2 Check your knowledge of measures of Spread

© 2003 Prentice-Hall, Inc. Comparing Standard Deviations Mean = 15.5 s = Data B Data A Mean = 15.5 s = Mean = 15.5 s = 4.57 Data C

© 2003 Prentice-Hall, Inc. Coefficient of Variation Measure of Relative Variation Always in Percentage (%) Shows Variation Relative to the Mean Used to Compare Two or More Sets of Data Measured in Different Units Sensitive to Outliers

© 2003 Prentice-Hall, Inc. Comparing Coefficient of Variation Stock A: Average price last year = $50 Standard deviation = $2 Stock B: Average price last year = $100 Standard deviation = $5 Coefficient of Variation: Stock A: Stock B:

© 2003 Prentice-Hall, Inc. Shape of a Distribution Describe How Data are Distributed Measures of Shape Symmetric or skewed Mean = Median =Mode Mean < Median < Mode Mode < Median < Mean Right-Skewed Left-SkewedSymmetric

© 2003 Prentice-Hall, Inc. Describing Distribution: Shape, Center and Spread Start by displaying the distribution of your data. You should describe: Symmetry vs. skewness * (Look at the tails) Single vs. multiple modes Possible outliers separated from the main body of the data.... But remember that not all distributions have simple shapes. Symmetry Skewness UnimodelBimodelmultimodalUniform Outliers 4.3 Recognize the important attributes of distribution shape in a histogram or stem-and- leaf display.

© 2003 Prentice-Hall, Inc. Identifying Distribution Shape When looking at the distribution of a quantitative variable, consider: Symmetry, How many modes it has, Whether it has any outliers or gaps. 4.4 Practice describing distribution shapes

© 2003 Prentice-Hall, Inc. Summary: Describing a Distribution You should always begin analyzing data by graphing the distributions of the variables. We describe the shape of the distribution of a quantitative variable by noting whether it is symmetric or skewed, unimodal, bimodal, or multimodal. Following the general rule that we first describe a pattern and then look for deviations from that pattern, we look at distributions to see if there are straggling outliers or gaps in the distribution shape. Each of these descriptions is quite general and vague, so people may disagree on the "close calls." Nevertheless, we usually begin describing data by summarizing the distribution shapes.

© 2003 Prentice-Hall, Inc. Exploratory Data Analysis Box-and-Whisker NOTE: Different from Basic Business Statistics Book! Graphical display of data using 5-number summary Median( ) Top of Whiskers Bottom of Whiskers

© 2003 Prentice-Hall, Inc. Distribution Shape & Box-and-Whisker Right-SkewedLeft-SkewedSymmetric

© 2003 Prentice-Hall, Inc. Box Plots Whenever we have a five-number summary of a (QUANTITIVE) variable we can produce a graph (display) called a box plot NOTE: Basic Business Statistics Book Does not Include Outliers! 1. High Outliers (Above Upper Fence) 2. Top of Whiskers 3. Q3 4. Median 5. Q1 6. Bottom of Whiskers 7. Low Outliers (Below Lower Fence)

© 2003 Prentice-Hall, Inc. Box Plot: Good way of Comparing Categories Category 1 EG Males Category 2 EG Females Outliers Whiskers do not extend further than 1.5 x IQR 5.4 Know how to make boxplots Top of Whiskers Bottom of Whiskers

© 2003 Prentice-Hall, Inc. 5 Steps to Create a Box Plot Draw Box Plots for the dataset: 7,8,25,11,13,5,6 We require to know: Median, Q1 (25% Percentile), Q3 (75% Percentile, Top of Whiskers, Bottom of Whiskers and Outliers Step 1Order Data Step 2Calculate Median N is odd so data point at position is the middle value therefore Median = 8

© 2003 Prentice-Hall, Inc. 5 Steps to Create a Box Plot (Cont). Step 3Calculate Q1 and Q3 Divide the above data set into Data set 1 and Data Set 2. Because n is odd include the median in both sets Data set 1 is used to calculate Q1 and Data set 1 is used to calculate Q3 Dataset 1 For Q1Dataset 2 For Q For Q1: N for dataset 1 is even so Q1 is the average of the two middle at position 2 and 3 Q1 = (6+7)/2 =6.5 (This is the same method used to calculate the median for a even number of data points)

© 2003 Prentice-Hall, Inc. 5 Steps to Create a Box Plot (Cont). For Q3: N for dataset 2 is even so Q3 is the average of the two middle at positions 2 and 3 Thus, Q3 = (11+13)/2 =12 Step 4 Find if there are any Outliers IQR= Q3-Q1 =5.5 Any data points above Upper Fence (UF) are outliers? UF =Q3+1.5*IQR =12+1.5*5.5=20.25 Any data points below Lower Fence (LF) are outliers? LF =Q1- 1.5*IQR = *5.5= Outliers

© 2003 Prentice-Hall, Inc. That 25 is the only data point that is an outlier Outliers Upper Fence (UF) =20.25 Lower Fence (LF) =-1.75 Bottom of Whisker is 5 Top of Whisker of is 13

© 2003 Prentice-Hall, Inc. Step 5: Create the Box Plot (Output from SPSS) Outlier Outlier = 25 Top of Whisker = 13 Bottom of Whisker = 5

© 2003 Prentice-Hall, Inc. Box Plots: Summary 1. Order Data 2. Find and Plot Median, Q1 and Q3 3. Calculate IQR 4. Calculate Upper Fence = Q * IQR 5. OUTLIERS: Plot numbers above the Upper Fence 6. Top of Whiskers: highest number excluding outliers 7. Calculate Lower Fence = Q *IQR 8. OUTLIERS: Plot numbers below the Lower Fence 9. Bottom of Whiskers: lowest number excluding outliers 7N = VAR Q3 Median Q1

© 2003 Prentice-Hall, Inc. Examples Calculate box plots for the following data points. 1. 5,20,15,3,7,16, ,22,22,23,25,2 3. 4,11,10,12, ,11,11,9,10,12,18

© 2003 Prentice-Hall, Inc. Box Plot can be used to compare two categories. Exam Results by Class Copied

© 2003 Prentice-Hall, Inc. When comparing groups with boxplots Compare the medians. Which group has a larger center? Compare the IQR's. Which group is more spread out? Compare the difference between the medians to the IQR's. On the scale of the IQR's are the medians very different? Check for possible outliers. Identify them if you can. Does knowing which values are outliers tell you anything about the data, the groups, or how they compare? Numeric comparisons of groups often consider the difference between the centers of the groups. This difference should be compared to the size of the spreads of the groups. Data such as the rat lifetimes data that have outliers are better summarized with medians and quartiles than with means and standard deviations.

© 2003 Prentice-Hall, Inc. Box plots explore the relationships among several groups. Easy to focus on center spread and symmetry Look for patterns of increasing spread and symmetry Lookout for outliers

© 2003 Prentice-Hall, Inc. Examples Do examples…..

© 2003 Prentice-Hall, Inc. The Empirical Rule For Most Data Sets, Roughly 68% of the Observations Fall Within 1 Standard Deviation Around the Mean Roughly 95% of the Observations Fall Within 2 Standard Deviations Around the Mean Roughly 99.7% of the Observations Fall Within 3 Standard Deviations Around the Mean

© 2003 Prentice-Hall, Inc. The Bienayme-Chebyshev Rule The Percentage of Observations Contained Within Distances of k Standard Deviations Around the Mean Must Be at Least Applies regardless of the shape of the data set At least 75% of the observations must be contained within distances of 2 standard deviations around the mean At least 88.89% of the observations must be contained within distances of 3 standard deviations around the mean At least 93.75% of the observations must be contained within distances of 4 standard deviations around the mean

© 2003 Prentice-Hall, Inc. Coefficient of Correlation Measures the Strength of the Linear Relationship between 2 Quantitative Variables

© 2003 Prentice-Hall, Inc. Features of Correlation Coefficient Unit Free Ranges between –1 and 1 The Closer to –1, the Stronger the Negative Linear Relationship The Closer to 1, the Stronger the Positive Linear Relationship The Closer to 0, the Weaker Any Linear Relationship

© 2003 Prentice-Hall, Inc. Scatter Plots of Data with Various Correlation Coefficients Y X Y X Y X Y X Y X r = -1 r = -.6r = 0 r =.6 r = 1

© 2003 Prentice-Hall, Inc. Scatter Plots Best way to see relationships between two Quantitative Variables If there is a relationship it is called an association. Scatter plots are the best way to look for associations

© 2003 Prentice-Hall, Inc. Scatter Plots Only Quantitative Variables can be used The following is the relationship between the Mango Taste Score and the Number of Weeks it grown for Weeks Grown on Tree Time taken to grow the best Mango Taste Score

© 2003 Prentice-Hall, Inc. Scatter Plots : DIRECTION and FORM Direction: +Ve Form: Linear (Straight Line) Direction: +Ve Form: Linear (Straight Line)

© 2003 Prentice-Hall, Inc. Form describes the shape Describe the Form if there is one. EG Straight, or arched or curved An Arch Taste Score

© 2003 Prentice-Hall, Inc. Scatter: The following Scatter Plots have no scatter The third thing to look for is scatter. These graphs have no Scatter because the dots follow each other in a line.

© 2003 Prentice-Hall, Inc. Scatter: The following Scatter Plots have scatter These graphs have Scatter because the dots are like a cloud. Always look for something you do not expect to find One Graph has an outlier which are very important unexpected points that may occur

© 2003 Prentice-Hall, Inc. How to Measure Scatter: Correlation Coefficient, r Sx = Sy =

© 2003 Prentice-Hall, Inc. Plot Chart and describe scatter plot. Calculate Correlation Coefficient These data points go in straight line with no scatter in a positive direction (positively correlated)

© 2003 Prentice-Hall, Inc. Example 2 For the following data, plot a scatter plot and describe it. Calculate Correlation Coefficient

© 2003 Prentice-Hall, Inc. Answer: Example 2 These data points go in straight line with some scatter in a negative direction (negatively correlated)

© 2003 Prentice-Hall, Inc. Example 3 For the following data, plot a scatter plot and describe it. Calculate Correlation Coefficient

© 2003 Prentice-Hall, Inc. Answer: Example 3 These data points are in an arched shape with a little scatter.

© 2003 Prentice-Hall, Inc. Example 4 For the following data, plot a scatter plot and describe it. Calculate Correlation Coefficient

© 2003 Prentice-Hall, Inc. Answer: Example 4 These data points have a lot of scatter and they are not correlated

© 2003 Prentice-Hall, Inc. Pitfalls in Numerical Descriptive Measures and Ethical Issues Data Analysis is Objective Should report the summary measures that best meet the assumptions about the data set Data Interpretation is Subjective Should be done in a fair, neutral and clear manner Ethical Issues Should document both good and bad results Presentation should be fair, objective and neutral Should not use inappropriate summary measures to distort the facts

© 2003 Prentice-Hall, Inc. Chapter Summary Described Measures of Central Tendency Mean, Median, Mode, Geometric Mean Discussed Quartiles Described Measures of Variation Range, Interquartile Range, Variance and Standard Deviation, Coefficient of Variation Illustrated Shape of Distribution Symmetric, Skewed, Using Box-and-Whisker Plots

© 2003 Prentice-Hall, Inc. Chapter Summary Described the Empirical Rule and the Bienayme-Chebyshev Rule Discussed Correlation Coefficient Addressed Pitfalls in Numerical Descriptive Measures and Ethical Issues (continued)