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BBA182 Applied Statistics Week 3 (1) Displaying data – shape of distributions Dr Susanne Hansen Saral Email: susanne.saral@okan.edu.tr

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Histogram of employee completion times Numerical data

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Numerical data Employee completion time Cumulative frequency

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Bar Chart – categorical data

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Describing distributions Once we have made a picture of our numerical data, the histogram, what can we say about it’s shape?

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Describing distributions – what to pay attention to! Pay attention to: its’ shape its’ center Its’ spread

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Describing the shape of distributions We describe the shape of a distribution in terms of: Modes Symmetry Gaps or outlying values

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Mode Does the distribution have one peak (mode) or several peaks (several modes)? Uni-modal: one mode Bi-modal: Two modes Multi-modal: More than two modes

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Symmetry If we can make a mirror image of the distribution, we have a symmetric distribution

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Skewed distribution The thinner parts of a distribution are called tails. A distribution is skewed, or asymmetric, if one tail stretches farther out on one side than on the other side of the center. A right skewed distribution has a tail that extends farther to the right. A left skewed distribution has a tail that extends farther to the left.

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Right skewed distributions Examples Employee salaries in a company Waiting times in a line

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Left skewed distributions Example Time to finish an exam Employees going home after work Customers going shopping in a shopping center on a Saturday

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Outliers Outliers are extreme data points in a data set that are not close to the majority of the other data points Example: Age of 10 people in a restaurant: 24 19 21 65 20 21 23 20 24 25

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Outliers If you are studying the personal wealth of Americans in 2010 and you have Bill Gates (Founder of Microsoft) in your sample. How would the personal wealth of Bill Gates affect the distribution of personal wealth of Americans in the sample?

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Outliers Outliers will affect the shape of a distribution:

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Outliers Outliers can affect almost every statistical method we use in Statistics. Therefore we need to look out for them. An outlier can be the most informative part in your data or it may just be an error. No matter what it is, you need to look at it critically and judge if it is important for our analysis.

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Graphs to Describe Time-Series Data A histogram can provide information about the distribution of a variable, but it cannot show any pattern of the data over time. Sometimes we need to analyze data over time. A graph of values against time is called a times series plot

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Graphs to Describe Time-Series Data A time-series plot is used to show the values of a variable ordered over time. Time is measured on the horizontal axis The variable of interest is measured on the vertical axis Used to monitor the evolution of a certain item of interest, such as evolution of the price of gas, annual interest rates, daily closing prices for shares of common stock, evolution of home prices in a certain region, exchange rates (Euro-TL, TL-$), etc.

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Line Chart (time series plot) One variable

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Line Chart (time series plot) Two variables

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Line Chart (time series plot) Two variables

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Presenting statistical charts and graphs When presenting data for an audience or a manager your charts and graphs MUST give as clear and accurate picture of the data as possible. The graphs and charts must be: Convincing Clear Truthful

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Manipulation of data Data can be manipulated in graphical techniques in such a way that they look more/less favorable than they are in reality. This gives misleading information about the data. You need to be critical whenever you are presented a graph, pie-chart, histogram, etc. You should also be careful not to construct misleading information with graphical techniques.

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Manipulation of data

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Identical data - different graph How is this possible?

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Manipulation of data

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Manipulation of data

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Manipulation of data What does this graph say about the data?

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Manipulation of data

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Histogram with equal interval width

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Data Presentation Errors Do not make a histogram of categorical data Unequal histogram interval widths Label the x-axis and y-axis clearly (identify variables clearly) Compressing or distorting the vertical axis Do not calculate numerical summaries of categorical data, such as code, telephone numbers, etc.

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Contingency table Class exercise A survey of the entering MBA students at a university in the US reported the following data on the gender of their students in their two MBA programs: What are the two variables under study?

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Contingency table How many students are surveyed? A) How many of all MBA students are women? B) How many of Two-year MBAs are women? C) How many of Evening MBAs are men? D) How many of all MBAs are men?

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Contingency table How many students are surveyed? Calculating the percent/probability of absolute frequencies: P = N = Total subjects surveyed = Subject of interest

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Contingency table in percent A) What percent of all MBA students are women? B) What percent of Two-year MBAs are women? C) What percent of Evening MBAs are men? D) What percent of all MBAs are men?

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Contingency table A) What percent of all MBA students are women? B) What percent of Two-year MBAs are women? C) What percent of Evening MBAs are men? D) What percent of all MBAs are men?

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Displaying categorical data -exercise Softdrink market share A local survey company conducted a survey on the consumption of soft drinks its region of operations. The results of the survey were summarized in the following pie-chart: A) Which soft-drink brand has the highest consumption? B) Is this a good method to display this data?

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Displaying categorical data -exercise Softdrink market share ( same data as in the preceding slide) A local survey company conducted a survey on the consumption of soft drinks its region of operations. The results of the survey were summarized in the following pie-chart: A) Compared to the pie chart in the preceding slide, which chart is better for displaying the relative proportion (per- cent) of market share B) Which chart gives the best visual picture of the data?

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In this situation-beverage marketshare Which of the two graphs gives the best picture of the data?

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Are there any outliers in the following data sets? If yes, explain: A) 15 21 20 54 18 17 22 22 B) 345 340 339 344 338 341 343 C) – 21 -23 -25 -18 -20 -63 -19 -22

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How would the outliers affect the mean of the data set? Would the outlier increase or decrease the mean of the respective datasets? A) 15 21 20 54 18 17 22 22 B) 345 340 339 344 338 341 343 C) – 21 -23 -25 -18 -20 -63 -19 -22