descriptive statistics i this module covers statistics commonly used to describe or summarize a set...

Descriptive Statistics I

This module covers statistics commonly used to describe or summarize a set of data, including measures of central tendency (mean, median, mode) and measures of variability (range, standard deviation, variance).

Author: Phillip E. Pfeifer

© 2011 Phillip E. Pfeifer and Management by the Numbers, Inc.

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STwo Kinds of Descriptive Statistics

MBTN | Management by the Numbers

• Measures of Central Tendency• Mean• Median• Mode

• Measures of Variability• Range (Maximum – Minimum)• Standard Deviation• Variance

This MBTN module covers these six statistical measures. The first three describe the “center” of a data set. The latter three describe the spread of a data set. With each definition, we identify and explain the Excel function one can use to calculate the measure.

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Definition

The Sample Mean = The arithmetic average of the set of data (number1 + number2 +… numbern) / n

Excel Function = Average(num1, num2, …, numn) - or - Average(first cell:last cell)

The Sample Mean


InsightIf you know the sample mean and the number of data values, you can multiply the two to calculate the total. This is one reason the sample mean is such a popular statistic.

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The Sample Mean


Question 1: What is the sample mean of the following set of daily vehicle sales for a week? M=2, T=8, W=4, R=13, F=2

Answer:

We know that sample mean = (number1 + number2 +… numbern) / n

Therefore, substituting in our values:

Sample Mean = (2 + 8 + 4 + 13 + 2) / 5 = 5.8

We can also quickly calculate the total by multiplying 5.8 average vehicles x 5 days = 29 vehicles for the week

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Definition

The Median = The median is the point in the middle. An equal number of values are above & below the median.

Note: If there are an even number of data values, the median is the average of the two middle values.

Excel Function = Median(num1, num2, …, numn) - or - Median(first cell:last cell)

The Median


InsightSorting the data makes it much easier to find the median.

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The Median


Question 1: What is the median of the following set of daily vehicle sales for a week? M=2, T=8, W=4, R=13, F=2

Answer:

We know that the median is the point in the middle of the sorted data set

Therefore, sorting our values:

Median = 2, 2, 4, 8, 13 = 4

Note that two values are below (2, 2) and two values are above (8, 13)

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The Median


Question 2: What would be the median if our data set consisted of vehicle sales for Tuesday - Friday? T=8, W=4, R=13, F=2

Answer:

We know that the median is the point in the middle of the sorted data set


Sorted Set = 2, 4, 8, 13

But, in this example, there are two points in the middle, 4 and 8. So take the average of the two points.

Median = (4 + 8) / 2 = 6

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EDefinition

The Mode = The Mode is the Value Occurring Most Often.

Note: If there are no repeated values, rather than say all values “tie” for most occurring we say the data do not have a mode.

Excel Function = Mode(num1, num2, …, numn) - or - Mode(first cell:last cell)

The Mode


Definitions

Unimodal = Where only one value occurs most often

Bimodal = Where two values tie for occurring most often

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EThe Mode


Question 1: What is the mode of the following set of daily vehicle sales for a week? M=2, T=8, W=4, R=13, F=2

Answer:

We know that the mode is the value that occurs most often


2, 2, 4, 8, 13

The mode is 2 as it occurs twice and the other three values occur only once.

We can also describe this data set as unimodal because there is only one mode.

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EThe Mode


Question 2: If the data set also included Saturday sales of 13 vehicles, what would be the mode of the 6-observation data set? M=2, T=8, W=4, R=13, F=2, S=13

Answer:

We know that the mode is the value that occurs most often


2, 2, 4, 8, 13, 13

The values 2 and 13 are both modes for this bimodal data set.

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Measures of Central Tendency


• Sample Mean• The Arithmetic Average

• Median• The Middle Value

• Mode• The Value Occurring Most Often

The ensemble of sample mean, median, and mode can tell you a lot about how the data values are distributed….as we shall now see.

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SSymmetry and Skewness


Definitions

If the data are unimodal and the mean, median, and mode are all equal, the data is said to be symmetric.

If the data are unimodal and the mean, median, and mode are all different, the data is said to be asymmetric.

Data is said to be skewed to the right where the data is characterized by a few large values and many small values. In this circumstance, the sample mean is normally greater than the median.

Data is said to be skewed to the left where the data is characterized by a few small values and many large values. In this circumstance, the sample mean is normally less than the median.

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Question 1: Describe the following data of car sales for a week in terms of symmetry and skewness. M=2, T=12, W=9, R=7, F=5, S=7

Answer:

First, let’s sort our values giving us: 2, 5, 7, 7, 9, 12

Mean = (2 + 5 + 7 + 7 + 9 + 12) / 6 = 7Median = 7 (middle value)Mode = 7 (occurs twice)

Therefore, the mean, median and mode are all equal, so the data set would be described as symmetric (not skewed)

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Question 2: Describe the following data of car sales for a week in terms of symmetry and skewness. M=2, T=21, W=9, R=2, F=3, S=5

Answer:

First, let’s sort our values giving us: 2, 2, 3, 5, 9, 21

Mean = (2 + 2 + 3 + 5 + 9 + 21) / 6 = 7Median = (3 + 5) / 2 = 4 (average of two middle values)Mode = 2 (occurs twice)

The mean, median and mode are not equal the data would be considered asymmetric. Because the mean, median and mode are not equal with the mode being less than the median which, in turn, is less than the sample mean---we say the data are skewed to the right.

InsightBusiness data sets are often skewed to the right (think of salaries, sales by customer, etc.)

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YMeasures of Variability



Many business decisions are based not only on averages, but also on variability around the average. Variability in temperature, for example, leads to higher heating/cooling cost. We turn now to three statistics that describe the spread of the data, e.g. measures of variability.

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GEDefinition

The Range = The difference between the maximum and minimum values in a data set.

Excel Function = Range(number1, number2, …, numbern) - or - Range(first cell:last cell)

The Range


Question 1: What is range of the following set of daily vehicle sales for a week? M=2, T=8, W=4, R=13, F=2

Answer:

We know that the range = Maximum - MinimumTherefore, substituting in our values:Range = 13 – 2 = 11

Note that the “range” is from 2 to 13, but the range of the data is 11.

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NSample Standard Deviation


InsightThink of the sample standard deviation as a measure of how variable the data are. If all the data take on the same value, the standard deviation will be zero.

Definition

The Sample Standard Deviation is the square root of the “average” squared distances of the points from the sample average.

(num1 – x )^2 + (num2 – x )^2 + … + (numn – x )^2 ^ (1/2)StdDev = n-1

Where x = sample average and n = number of data points in the data set

Excel 2010 Function = stdev.s(num1, num2, …, numn)Excel 2007 Function = stdev(num1, num2, …, numn)

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Question 1: What is the sample standard deviation of the following set of daily vehicle sales for a week? M=2, T=8, W=4, R=13, F=2

Answer:

We know that sample mean = (number1 + number2 +… numbern) / n

Therefore, substituting in our values:

Sample Mean = (2 + 8 + 4 + 13 + 2) / 5 = 5.8

Then continuing our calculation for the sample standard deviation…

Sum of Squared differences = (2 – 5.8)^2 + (8 - 5.8)^2 … + (2 - 5.8)^2 = 88.8Std Dev = (88.8 / (5 – 1))^.5 = 4.71

Doing just one by hand will quickly demonstrate why Excel is such a valuable tool for statistics!

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Insight

The sample standard deviation is a better measure of variability than the range because it uses all the data points (and for other technical reasons we will not get into.)

To find a sample standard deviation, you will almost always use Excel….even if there are few data points.

If there are lots of data points with a unimodal, symmetric (bell-shaped) distribution, a rough rule of thumb says that 68% of the values will fall within one standard deviation of the sample average.

Using our previous example where the sample average = 5.8 and the standard deviation = 4.71 (and presuming a bell-shaped distribution – not the case), our rule of thumb would then say that we would expect 68% of the values to fall between 5.8 – 4.71 and 5.8 + 4.71 (or between approx. 1.1 and 10.5)

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Sample Variance


InsightIf this looks familiar, it should! Calculating sample variance requires all the steps in calculating sample standard deviation..except the final square root. Therefore, variance also equals StdDev ^ 2.

Definition

The Sample Variance is the “average” squared distances of the points from the sample average (also the square of the standard deviation).

(num1 – x )^2 + (num2 – x)^2 +…+ (numn – x)^2

Sample Variance = n - 1

Where x = sample average and n = number of data points in the data set

Excel 2010 Function = var.s(num1, num2, …, numn)Excel 2007 Function = var(num1, num2, …, numn)

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Sample Variance


Question 1: What is the sample variance of the following set of daily vehicle sales for a week? M=2, T=8, W=4, R=13, F=2

Answer:

Sample Mean = (2 + 8 + 4 + 13 + 2) / 5 = 5.8

Then continuing our calculation for the sample variance…

Squares of the differences = (2 – 5.8)^2 + (8 - 5.8)^2 … + (2 - 5.8)^2 = 88.8Variance = (88.8 / (5 – 1)) = 22.2

InsightSince the sample variance is the square of the sample standard deviation, if you know one you can easily calculate the other. Generally, the standard deviation is much easier to interpret, in part, because it has the same units as the data. (e.g. the 4.71 sample standard calculated earlier is 4.71 cars. The 22.2 is cars^2.)

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Descriptive Statistics


• Measures of Central Tendency• Mean• Median• Mode


This completes our introduction to the six descriptive statistics listed above. What follows are a couple of slides that show how these statistics behave if you multiply the data by a constant “b” and add another constant “a”. This is called a linear transformation. The transformations used to convert pounds to kilograms, feet to miles, and millions to billions are all examples of linear transformations.

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TADescriptive Statistics for Transformed Data


Let X represent the original data.Let Y = a + b * X be the transformed data.

Sample mean(Y) = a + b * Sample Mean(X)Median(Y) = a + b * Median(X)Mode(Y) = a + b * Mode(X)

InsightThe mean, median, and mode all behave in the logical way for the linearly transformed data. Thus, if the median temperature was 68 degrees Fahrenheit, the median temperature (if calculated using the same data expressed in degrees Celsius) would be (5/9) * (68-32) = 20 degrees Celsius. This is true because the transformation of Fahrenheit to Celsius is linear…and because of the way the three statistics behave.

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TADescriptive Statistics for Transformed Data


Let X represent the original data.Let Y = a + b * X be the transformed data.

Range(Y) = abs(b) * Range(X)Standard Deviation(Y) = abs(b) * Standard Deviation(X)Variance(Y) = b^2 * Variance(X)

InsightSince range, standard deviation, and variance all measure variability, it might come as no surprise that adding a constant to the data does NOT affect these three statistics. Multiplying the data by a constant, however, multiplies the range and standard deviation by the absolute value of the constant and multiplies the variance by the constant squared. Thus if the standard deviation of temperatures was 10 degrees Fahrenheit, the standard deviation of the same data would be (5/9)*10 or 50/9 in degrees Celsius.

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Any Introductory Statistics Book such as Introductory Statistics (9th Edition), Neil. A. Weiss, Pearson Publishing, 2010.

Two-Variable Descriptive Statistics (advanced MBTN module – coming soon). This module provides further insight into statistics including correlation and regression.

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Descriptive Statistics - Further Reference


descriptive statistics i this module covers statistics commonly used to describe or summarize a set...

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