# Deciles in a Data Set: Definition, Formula & Examples

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• 0:04 Grouping Data
• 0:58 Quantiles and Deciles
• 3:42 Frequency Distribution Data
• 5:35 Lesson Summary
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Lesson Transcript
Instructor: Bob Bruner

Bob is a software professional with 24 years in the industry. He has a bachelor's degree in Geology, and also has extensive experience in the Oil and Gas industry.

When faced with large amounts of data, it's often useful to group the data prior to further analysis. Given that we work with a base ten number system, it's logical that we might want to use divisions of 10, or deciles, as a basic form of numerical grouping.

## Grouping Data

Let's say that you were working as a sales manager. If you were a sales manager, it would be useful to know the average and median values of all sales made to your customers. But if you also found out that 80% of your sales came from just the top 20% of your customers, that would be much more informative - and might even change the way you approached your future sales campaigns.

When we analyze data it is useful to first bring structure to the raw data values, particularly when working with larger data sets. One way to bring structure is to first group the values and then look for patterns and anomalies in those groups. In some cases you may want to group data based on prior information, or known categories. For example, if your company was organized into individual sales regions, it would be logical to view your data accordingly. But in cases where you don't have prior knowledge or are just starting an analysis, grouping data is often done numerically.

## Quantiles and Deciles

A quantile is a set of values used to divide any frequency distribution into equal groups. This means that each group will contain the same portion of the total population, and the resulting frequency distribution will be flat. A decile is a specific type of quantile that arranges data into 10 equal parts. In order to create deciles we must actually derive 9 specific numbers, or cut points, that define where these deciles begin and end.

Suppose we're given a set of unstructured data points. The very first thing that we must do to find our 9 cut points is to order the data sequentially. Let's use an example based on 23 random numbers valued from 20 to 78. If our raw numbers are:

24, 32, 27, 32, 23, 62, 45, 77, 60, 63, 36, 54, 57, 36, 72, 55, 51, 32, 56, 33, 42, 55, 30

Then our sequentially ordered data looks like this:

23, 24, 27, 30, 32, 32, 32, 33, 36, 36, 42, 45, 51, 54, 55, 55, 56, 57, 60, 62, 63, 72, 77

If i = (1, 2, 3, ..., 9), then the nine numbers we require can be represented as D1, D2, ... , D9.

The cut point values are calculated using the number of observations in the data set, n, and the following formula:

The resulting number defines the place value, along the sequential data set, at which any decile begins or ends.

Suppose we wanted to find the value for the 5th point. If we plug in our values, D5 calculates to 12.0. In this case, because there is no fractional value in the computation, we can use the 12th term from our ordered list directly. Counting across the ordered list, the associated look up value for D5 at the 12th location is 45.

Oftentimes the calculation results in a value that has a decimal component. For example, D9 calculates to 21.6. In this case we first locate the 21st value, 63, and then add 0.6 of the difference between that value and the next higher value, which in our case is 72.

The value for D9 = 63 + 0.6 * (72 - 63) = 68.4

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