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BBA182 Applied Statistics Week 3 (2) Using numerical data to describe data Dr Susanne Hansen Saral Email: susanne.saral@okan.edu.tr

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Using numerical measures to describe data «Is the data in the sample centered or located around a specific value?» First question that business people, economists, corporate executives, etc. ask when presented with sample data.

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Using numerical measures to describe data The histogram gives an idea whether the data is centered around a specific value. The histogram provides a visual picture of how the data is distributed (symmetric, skewed, etc.)

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Is the data centered around a specific value?

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Numerical measures to describe data

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Measures of the center of the data set

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Mean Population mean, The mean is the most common measure of the center of a data set For a population of N values:

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Mean Sample Mean, For a sample of n values:

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The Mean symmetry and unimodal distribution When we have a symmetric distribution with one Mode, then the mean represents the middle value in a data set.

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Mean The most common measure for the center of a data set Affected by extreme values (outliers)

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Mean The most common measure for the center of a data set Affected by extreme values (outliers)

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Skewed distribution An outlier will distort the picture of the data. It will inflate or deflate the mean, depending on the value of the outlier This creates a skewed distribution. In this case we may want to use a different measure of the data center

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Median In an ordered list of data, the median is the “middle” number (50% above, 50% below) Not affected by outliers

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Finding the Median The location of the median: If the number of values is odd (uneven), the median is the middle number - 17 6 25 -5 13 9 33 For this data set: -17 -5 6 9 13 25 33

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Finding the Median The location of the median: If the number of values is even, the median is the two middle numbers divided by 2

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Finding the median Determine the median of the following data set: 17 5 3 11 12 8 25 3

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Finding the median Determine the median of the following data set: 17 5 3 11 12 8 25 3 3 3 5 8 11 12 17 25 Median: 8 +11 = 19/ 2 = 9.5

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Mode Value that occurs most often in the data set Not affected by outliers Used for either numerical or categorical data There may be no mode There may be several modes, uni-modal, bi-modal, multimodal

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Measures of the center summary data Five houses on a hill by the beach

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Measures of the center summary data What is the mean house price? What is the median house price? What is the modal house price?

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Mean: ($3,000,000/5) = $600,000 Median: middle value of ranked data = $300,000 Mode: most frequent house price = $100,000

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When is which measure of the center the “best”? Mean is generally used, unless outliers exist. If there are outliers the mean does not represent the center well. Then median is used when outliers exist in the data set. Example: Median home prices may be reported for a region – less sensitive to outliers

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Shape of a Distribution Describe the shape of a distribution Describes how data is distributed The presence or not of outliers in a data set, influence the shape of a distribution Symmetric or skewed

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Histogram of annual salaries (in $) for a sample of U.S. marketing managers: Describe the shape of this histogram (of the distribution) Without doing calculations. Do you expect the mean salary to be higher or lower than the median salary?

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Class exercise Eleven economists were asked to predict the percentage growth in the Consumer Price Index over the next year. Their forecasts were as follows: 3.6 3.1 3.9 3.7 3.5 1.0 3.7 3.4 3.0 3.7 3.4 Compute the mean, median and the mode Are there any outliers in the data set that may influence the value of the mean? If there are outliers, how do they affect the shape of the data distribution?

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Solution to class exercise Mean: 36/11 = 3.27 rounded up to 3.3 Median: 3.5 Mode: 3.7 Outlier: 1.0 How does the outlier affect the shape of the distribution? It decreases the average of the data set and distorts the picture of the histogram. The shape is skewed to the left.

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Measures of variability The three measures of data center do not provide complete and sufficient description of the data. Next to knowing how data is located around a specific value (mean, median or mode), we need information on how far the data is spread from that specific value, most often from the mean. The measure of variability will provide us with this information.

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Measures of Variability

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Quartiles Quartiles are descriptive measures that separate large data set into four quarters. The first quartile ( separates approximately the smallest 25 % of the data from the remaining largest 75 % of the data. The second quartile (), is the median, which separates the data set into two identical halves. The third quartile ( separates approximately the smallest 75 % of the data from the remaining largest 25 % of the data

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Quartiles

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How to calculate quartiles manually

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Quartiles

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Quartiles

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Quartiles and Enron case In the Enron data we had 60 data points. There are 30 values to right and 30 values to left side of the median (: ( = -$1.68 (between15th and 16th data points) - 75 % of the data is larger than -$ 1.68 ( = -$ 0.19 median (between 30th and 31st points) - 50 % of the data is smaller than -$.19 and 50 % of the data is larger than -$.19 . ( = $2.14 (between 45th and 46th data pots) - 25 % of the data is larger than $2.14

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Range Simplest measure of variation Difference between the largest and the smallest observations:

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Range – Example Enron case Range = Maximum value – minimum value Enron data range = $21.06 – (-$17.75) = $ 38.81

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Disadvantages of the Range Ignores the way in which data is distributed

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Disadvantages of the Range Sensitive to outliers

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Range: short-comings as a good measure for variability Because the range does not provide us with a lot of information about the spread of the data it is not a very good measure for variability.

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Interquartile Range We can eliminate some outlier problems by using the interquartile range and concentrate on the middle 50 % of the data in the data set Eliminate high- and low-valued observations and calculate the range of the middle 50% of the data Q1 Q3 Interquartile range The Interquartile range, IQR =

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Interquartile Range The interquartile range (IQR) measures the spread of the data in the middle 50% of the data set Defined as the difference between the observation at the third quartile and the observation at the first quartile IQR = Q3 - Q1

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Interquartile Range Raw data: 6 8 10 12 14 9 11 7 13 11 n = 10 Ranked data: 6 7 8 9 10 11 11 12 13 14 1. Quartile: 7.75 3. Quartile: 12.25 IQR = Q3 – Q1 = 12.25 – 7.75 = 4.5 Q1: 7.75 Q3: 12.25

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Enron data: Interquartile range Interquartile range: IQR : $2.14 – (-$ 1.68) = $ 3.82 The middle 50 % of the Enron data has a spread of $ 3.82 compared to the range of $ 38. 81!