GB701 Unit 2 Descriptive Summary Exercise Example

Two Populations in One Histogram: A Unit 2 Descriptive Summary of Quote-to-Order Time

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Purdue University Global

GB701: Business Statistics

Unit 2 Descriptive Summary Exercise

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[Date]

The distributor, its accounts and every figure are composite and illustrative; the methods and sources cited are real.

The Variable

Quote-to-order time is each account's average number of days from the date a quote is issued to the date the customer's purchase order is received, across all quotes that converted to orders in the twelve months ending in June. Quotes that never converted are excluded, since they have no order date; their count is recorded separately. The data come from the quoting system's conversion report, matched to the customer master file for order type. An account is classed as special-order if more than half of its quoted lines in the year were items not held in stock.

Defining the variable this precisely matters because "days to order" could mean several things: days from the first quote in a revision series or from the last, days to the first order or to the final one, calendar days or business days. This summary uses calendar days from the final revised quote to the purchase order. Tukey (1977) urged analysts to settle what a number measures before computing anything from it, and every figure below inherits these choices.

Table 1: Summary for All Accounts

Count, 1,240. Mean, 8.55 days. Five percent trimmed mean, 8.1. Median, 7.5. Standard deviation, 5.01. Interquartile range, 5.6, from a first quartile of 5.0 to a third quartile of 10.6. Skewness, 1.62. Excess kurtosis, 3.89. Minimum, 1.5. Maximum, 35.9.

Why the Median Leads

The distribution is right-skewed: most accounts convert within about two weeks, and a long tail stretches past a month. Skewness of 1.62 and excess kurtosis of 3.89 confirm what the histogram shows. In a skewed distribution the mean is pulled toward the tail, here about a day above the median, so the median better represents the typical account. The interquartile range, which ignores the tails, describes the spread of the middle half more honestly than the standard deviation, which the tail inflates. The summary therefore reports the median and interquartile range first, with the mean and standard deviation alongside for readers who need them for later calculations.

What the Trimmed Mean Adds

The 5 percent trimmed mean, 8.1 days, removes the fastest and slowest 5 percent of accounts before averaging. It sits between the mean and the median, which suggests the tail explains much of the distance between them, though not all of it. Wilcox (2017) recommends trimmed means for skewed business and behavioral data because they are less sensitive to extreme values while using more of the data than the median does.

Figure 1: Panels Instead of a Pooled Chart

The sales team's standard chart is a single histogram of all accounts in five-day bins. It shows one lump with a long right tail, and readers tend to see one population with some slow outliers.

Figure 1 replaces it with two histograms stacked on a shared axis, one for the 891 stock-order accounts and one for the 349 special-order accounts, both in two-day bins. The stock panel peaks around five to seven days and falls off quickly. The special-order panel is flatter and centered near ten days, with most of the long tail. Shown separately, the "slow outliers" turn out to be the bulk of a different population.

The slow accounts were not outliers from one distribution; they were the middle of a second one.

Figure 2: Box Plots Side by Side

Figure 2 places box plots for the two groups next to each other. The boxes barely overlap: the stock group's third quartile, 9.2 days, sits close to the special-order group's first quartile, 7.7 days. Cleveland (1993) argues that displays should be chosen so that the comparisons a reader needs to make are easy to see, and the side-by-side box plots make the difference between the groups visible in seconds.

Table 2: Summary by Order Type

Stock-order accounts, 891. Mean 7.28 days, standard deviation 3.84, median 6.4, interquartile range 4.7 (4.5 to 9.2), maximum 28.6, skewness 1.34.

Special-order accounts, 349. Mean 11.80 days, standard deviation 6.06, median 10.5, interquartile range 7.2 (7.7 to 14.9), maximum 35.9, skewness 1.28.

The special-order group is both slower and more variable. Its interquartile range is about half again as wide as the stock group's.

What the Display Changes for Readers

The split display changes what the sales team will discuss. With the pooled chart, conversations about slow accounts focused on individual customers who seemed unusually hesitant. With the panels, the conversation shifts to why special-order quotes take longer as a group, which points toward lead times from manufacturers, engineering approvals at the customer's site and the number of revisions a special-order quote goes through.

Limits Stated Plainly

Three limits apply. Averaging each account's quotes hides variation within accounts, and an account with many quotes counts the same as one with few. The special-order classification uses a 50 percent threshold, and accounts near the line could fall either way. And excluding unconverted quotes means the summary describes how fast customers order when they do, not how often they order at all.

References

Cleveland, W. S. (1993). Visualizing data. Hobart Press.

Tukey, J. W. (1977). Exploratory data analysis. Addison-Wesley.

Wilcox, R. R. (2017). Introduction to robust estimation and hypothesis testing (4th ed.). Academic Press.