Chapter 2 – Descriptive Statistics Tabular and Graphical Presentations

Chapter Outline Summarize Qualitative Data Summarize Quantitative Data Frequency Distribution Bar Charts and Pie Charts Summarize Quantitative Data Histogram Cumulative Distributions Crosstabulations Scatter Diagrams

A Note An important aspect of statistics is to present the data in an informative way so as to reveal any patterns in the data (no pattern is a pattern!). Different types of data require different summarization methods and statistical analyses.

Summarize Qualitative Data Check out the following data. What pattern can you detect from the raw data?

Summarize Qualitative Data Frequency Distribution The raw data in the previous table does not provide any meaningful information ( like any pattern) directly. For qualitative data, we can summarize and present the raw data with ‘Frequency Distribution’. A frequency distribution is a tabular summary of data showing the number (frequency) of items in each nonoverlapping class. Please refer to the Excel demonstration ( Chapter 2) on how to construct the frequency distribution for the data in table 2.1. The outcome is shown on the next slide.

Frequency Distribution for Data in Table 2.1

Relative Frequency To obtain relative frequency, simply divide the frequency of each class by the total number of observations (n). For the data in Table 2.1, n equals 50. 15/50=0.3

Bar Charts and Pie Charts A frequency distribution is often presented in a graph (a bar chart or a pie chart) to communicate information visually. Please refer to the Excel demonstration ( Chapter 2) on how to create a bar chart and a pie chart for the frequency distribution from previous slide. Both charts indicate that the most popular network evening news is on NBC.

Summarize Quantitative Data Check out the following data. Can you quickly decide how many classes there should be in the construction of a frequency distribution?

Summarize Quantitative Data Frequency Distribution Different from the qualitative data in Table 2.1, the quantitative data in Table 2.2 do not indicate the number of classes straightforwardly. Apply the following procedure to construct a frequency distribution for quantitative data. Determine the number of non-overlapping classes; Determine the class width; Determine the class limits; Count the item numbers in each class.

Summarize Quantitative Data Frequency Distribution Step one – Determine the number of non-overlapping classes As a guidance, you can use the ‘2 to the power of k’ rule. That is, to find the smallest integer (k) such that 2k  n ( n is the sample size). Applying the rule to the data in Table 2.2, we find k = 6 since 26=64 ( n=50). Thus, we set the # of classes as 6. (Note that it is only a suggestion, not an absolute rule.) Empirically speaking, the # of classes is between 5 and 20.

Summarize Quantitative Data Frequency Distribution Step two – Determine the class width Use equal class width to avoid misinterpretation Approximately, class width = For the data in Table 2.2, class width = (120-61)/6= 9.96. We can round it up to 10, which is a much more convenient value to work with for class width.

Summarize Quantitative Data Frequency Distribution Step three – Determine the class limits Class limits should be set so that each data point belongs to one and only one class, and no data point is left out. Similar to class width, class limits can use values that are convenient to work with. In our example, the smallest value is 61 and the class width is set as 10. So, the lowest class can be set as 61 – 70. Note that the class width is calculated as 70-61+1=10.

Summarize Quantitative Data Frequency Distribution Step four – count the # of items in each class For the data in Table 2.2, the frequency distribution is constructed as follows: Please refer to the Excel demonstration ( Chapter 2) on how to construct the frequency distribution for the data in table 2.2.

Relative Frequency Example: Monthly Sales Volume of 50 Starbucks Stores 3/50=0.06

Interpretation of Frequency Distribution The frequency distribution of monthly sales volume of 50 Starbucks stores in NYC reveals that 39 stores generated an average monthly sales in 2012 between $81,000 and $110,000. 4% of the sample stores had an average monthly sales no more than $70,000. 6% of the sample stores had an average monthly sales $111,000 or more.

Histogram Like a bar chart, a histogram is a graphical presentation of frequency distribution. The height of a rectangle ( a bar) drawn above each class interval corresponds to that class’ frequency or relative frequency. Unlike a bar chart, a histogram has no gap between rectangles of adjacent classes. Please refer to the Excel demonstration ( Chapter 2) on how to create a histogram for the frequency distribution of Sales volume of Starbucks stores.

Monthly Sales Volume of 50 Starbucks Stores in NYC Histogram Monthly Sales Volume of 50 Starbucks Stores in NYC

Histogram Skewness – the lack of symmetry. Symmetric distribution, such as height or weight of human population. .05 .10 .15 .20 .25 .30 .35 Relative Frequency

Histogram Negative Skewness – a longer tail to the left. An example: exam scores Relative Frequency .05 .10 .15 .20 .25 .30 .35

Histogram Positive Skewness – a longer tail to the right. An example: home values Relative Frequency .05 .10 .15 .20 .25 .30 .35

Cumulative Distributions Cumulative frequency distribution – shows the # of items with values less than or equal to the upper limit of each class. Cumulative relative frequency distribution – shows the proportion (percentage) of items with values less than or equal to the upper limit of each class.

Cumulative Distributions Monthly sales volume of 50 Starbucks stores      2+6+11=19 19/50=0.38 

Crosstabulations and Scatter Diagrams So far, we have studies the methods of summarizing the data of one variable at a time. In business, it is important to understand the relationships among different variables. For instance, the relationship between sales volume and expenditure on advertisement. Crosstabulations and scatter diagrams are two methods of descriptive statistics, which are used to summarize the data to reveal the relationship of two variables.

Crosstabulations A crosstabulation is a tabular summary of data for two variables. The two variables can be either qualitative or quantitative or one of each. The left and top margin labels show the classes for the two variables.

Colonial Log Split A-Frame Crosstabulations Example: Finger Lakes Homes The number of Finger Lakes homes sold for each style and price for the past two years is shown below. quantitative variable categorical variable Home Style Price Range Colonial Log Split A-Frame Total 18 6 19 12 55 45 < $200,000 > $200,000 12 14 16 3 Total 30 20 35 15 100

Crosstabulations Insights Gained from Preceding Crosstabulation Example: Finger Lakes Homes Insights Gained from Preceding Crosstabulation The greatest number of homes (19) in the sample are a split-level style and priced at less than $200,000. Only three homes in the sample are an A-Frame style and priced at $200,000 or more.

Crosstabulation Insights Gained from Preceding Crosstabulation Example: Finger Lakes Homes Insights Gained from Preceding Crosstabulation The greatest number of homes (19) in the sample are a split-level style and priced at less than $200,000. Only three homes in the sample are an A-Frame style and priced at $200,000 or more.

Crosstabulations Example: Finger Lakes Homes Home Style Price Range Frequency distribution for the price range variable Example: Finger Lakes Homes Home Style Price Range Colonial Log Split A-Frame Total 18 6 19 12 55 45 < $200,000 > $200,000 12 14 16 3 Total 30 20 35 15 100 Frequency distribution for the home style variable

Crosstabulations: Simpson’s Paradox Data in two or more crosstabulations are often aggregated to produce a summary crosstabulation. We must be careful in drawing conclusions about the relationship between the two variables in the aggregated crosstabulation. In some cases the conclusions based upon an aggregated crosstabulation can be completely reversed if we look at the unaggregated data. The reversal of conclusions based on aggregate and unaggregated data is called Simpson’s paradox.

Scatter Diagrams A scatter diagram is a graphical presentation of the relationship between two quantitative variables. One variable is shown on the horizontal axis and the other variable is shown on the vertical axis. The general pattern of the plotted points suggests the overall relationship between the variables. A trendline provides a linear approximation of the relationship.

Scatter Diagrams A Positive Relationship y x

Scatter Diagrams A Negative Relationship y x

Scatter Diagrams No Relationship y x

Scatter Diagrams An example Is there a relationship between gas prices and stock prices? For the variable – gas price, let us use the data of the U.S. retail gas price; For the variable – stock prices, let us use the data of the S&P 500 Index ( ticker symbol – SPY); Weekly data for both variables. The data are shown in the next slide.

Data of U.S. Retail Gas Price and S&P 500 Proxy Price (SPY)

Scatter Diagrams The relationship between gas prices and stock prices

Scatter Diagrams The relationship between gas prices and stock prices The plots in the previous scatter diagram indicate a positive relationship between U.S. retail gas price and the value of SPY. The relationship is sketchy. When gas price is high, the S&P 500 Index tend to be high. We need to be cautious in drawing conclusion from a scatter diagram. In the example, there are only 10 data points. Much more data are required to rigorously examine the relationship between gas price and stock prices.