descriptive statistics i - management by the numbers statistics i this module covers statistics...

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.

TW

O K

IND

S O

F D

ES

CR

IPT

IVE S

TA

TIS

TIC

S

2

Two 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.

TH

E S

AM

PL

E M

EA

N

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)

3

The Sample Mean


Insight

If 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.

TH

E S

AM

PL

E M

EA

N

4

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

TH

E M

ED

IAN

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)

5

The Median


Insight

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

TH

E M

ED

IAN

6

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)

TH

E M

ED

IAN

7

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

TH

E M

OD

E Definition

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)

8

The Mode


Definitions

Unimodal = Where only one value occurs most often

Bimodal = Where two values tie for occurring most often

TH

E M

OD

E

9

The 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.

TH

E M

OD

E

10

The 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.

ME

AS

UR

ES O

F C

EN

TR

AL T

EN

DE

NC

Y

11

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.

SY

MM

ET

RY A

ND

SK

EW

NE

SS

12

Symmetry 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.

SY

MM

ET

RY A

ND

SK

EW

NE

SS

13



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 = 7

Median = 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)

SY

MM

ET

RY A

ND

SK

EW

NE

SS

14



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 = 7

Median = (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.

Insight

Business data sets are often skewed to the right (think of salaries,

sales by customer, etc.)

ME

AS

UR

ES O

F V

AR

IAB

ILIT

Y

15

Measures of Variability





• Variance

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.

TH

E R

AN

GE Definition

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)

16

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 - Minimum


Range = 13 – 2 = 11

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

SA

MP

LE S

TA

ND

AR

D D

EV

IAT

ION

17

Sample Standard Deviation


Insight

Think 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)

SA

MP

LE S

TA

ND

AR

D D

EV

IAT

ION

18



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


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 = 88.8

Std 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!

SA

MP

LE S

TA

ND

AR

D D

EV

IAT

ION

19



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)

SA

MP

LE V

AR

IAN

CE

20

Sample Variance


Insight

If 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)

SA

MP

LE V

AR

IAN

CE

21

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 = 88.8

Variance = (88.8 / (5 – 1)) = 22.2

Insight Since 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.)

DE

SC

RIP

TIV

E S

TA

TIS

TIC

S

22

Descriptive Statistics


• Measures of Central Tendency

• Mean

• Median

• Mode




• Variance

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.

DE

SC

RIP

TIV

E S

TA

TIS

TIC

S F

OR

TR

AN

SF

OR

ME

D D

AT

A

23

Descriptive 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)

Insight

The 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.

DE

SC

RIP

TIV

E S

TA

TIS

TIC

S F

OR

TR

AN

SF

OR

ME

D D

AT

A

24

Descriptive 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)

Insight

Since 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.

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.

DE

SC

RIP

TIV

E S

TA

TIS

TIC

S –

FU

RT

HE

R R

EF

ER

EN

CE

25

Descriptive Statistics - Further Reference


descriptive statistics i - management by the numbers statistics i this module covers statistics...

Documents