Data Analysis

Endurance3

cf_MAT510_w3_ch3.pptx

Home >Business & Finance homework help >Management homework help >Data Analysis

Business Statistics: A Decision-Making Approach

Tenth Edition

Chapter 3

Describing Data Using Numerical Measures

Slide - ‹#›

If this PowerPoint presentation contains mathematical equations, you may need to check that your computer has the following installed:

1) MathType Plugin

2) Math Player (free versions available)

3) NVDA Reader (free versions available)

Objectives

Compute the mean, median, mode, and weighted mean for a set of data and understand what these values represent.

Construct a box and whisker graph and interpret it.

Compute the range, interquartile range, variance, and standard deviation and know what these values mean.

Compute a z score and the coefficient of variation and understand how they are applied in decision-making situations.

Understand the Empirical Rule and Tchebysheff’s Theorem.

Slide - ‹#›

Section 3.1 Measures of Center and Location

Slide - ‹#›

Parameter and Statistic

Parameter

A measure computed from the entire population

As long as the population does not change, the value of the parameter will not change.

Statistic

A measure computed from a sample that has been selected from a population

The value of the statistic will depend on which sample is selected.

Slide - ‹#›

Describing Data – What to Look For

Measuring the Center

Data for a variable of interest forms a distribution

We need to be able to describe the center using a numerical measure.

Measuring the Spread

Values for a variable of interest will take on different values –the data are spread out around the center.

We need to be able to measure the spread.

Slide - ‹#›

Data Distributions with Different Centers

Slide - ‹#›

Data Distributions with Different Spreads

Slide - ‹#›

Descriptive Measures of the Center

Mean

The arithmetic average of the data

Median

The midpoint of the data

Mode

The data value occurring most frequently

Slide - ‹#›

Population Mean (Parameter)

µ - Population mean (pronounced mu)

N - Population size

individual value of variable x

The average for all values in the population computed by dividing the sum of all values by the population size

Slide - ‹#›

Population Mean - Example

Each day, a local hospital counts the number of patients that are in the hospital. This is called the hospital census. The hospital is interested in only the census data for the past 10 days. This is the population of interest. The data are shown below:

216	255	330	254	348
317	292	267	310	295

Slide - ‹#›

The Population Mean - Graphic Representation

Hospital Census – Population Data

Slide - ‹#›

Population Mean – Example (1 of 2)

Table 3.1 San Carlo Hotel Data

N = 8 Weeks

Week	Rooms Rented	Revenue	Complaints
1	22	$1,870	0
2	13	$1,590	2
3	10	$1,760	1
4	16	$2,345	0
5	23	$4,563	2
6	13	$1,630	1
7	11	$2,156	0
8	13	$1,756	0

= Total number of rooms rented

= Total dollar revenue from the room rentals

= number of customer complaints that came from guests each Sunday

Slide - ‹#›

Population Mean – Example (2 of 2)

Week	Rooms Rented
1	22
2	13
3	10
4	16
5	23
6	13
7	11
8	13

Calculate the Population Mean for the Variable, Rooms Rented

Slide - ‹#›

Mean – The Balance Point

Slide - ‹#›

Property of the Mean

Table 3.2 Deviations Around the Mean Using Hotel Data

Slide - ‹#›

Sample Mean (Statistic)

- Sample mean (x-Bar)

n - Sample size

The average for all values in the sample computed by dividing the sum of all sample values by the sample size

Slide - ‹#›

Sample Mean - Example

Number of Sales Made By a Sales Representative

n = 10 sample of ten days during the past three years

20	5	10	40	50
15	30	20	20	90

Raw Data

X-Bar

Slide - ‹#›

Sample Mean – The Balance Point

Sales Example

Slide - ‹#›

Central Property of the Mean

Sales Example

Slide - ‹#›

When Extreme Values Are Present Impact on the Mean -Sales Example

Original Data

New Data (one value changed)

Conclusion: The Mean is Sensitive to Extreme Values.

Slide - ‹#›

Impact of Extreme Values on the Mean –Accounting Graduates Starting Salaries - Example

Slide - ‹#›

Impact of Extreme Values on the Mean –Accounting Graduates Starting Salaries Example Using Excel 2016

Slide - ‹#›

Characteristics of the Mean

Uses all the Data

Sensitive to Extreme Values

The Balance Point

Sum of Deviations from the Mean Equals Zero

Slide - ‹#›

Median (1 of 2)

The median is a center value that divides a data array into two halves (Md).

Data Array

Data that have been arranged in numerical order

Median Index

i = The index of the point in the data set corresponding to the median value

n = Sample size

Slide - ‹#›

Median (2 of 2)

In an ordered array (lowest to highest), the median is the “middle” number, i.e., the number that splits the distribution in half numerically.

50% of the data is above the median, 50% is below

Represented as Md

The median is not affected by extreme values.

Slide - ‹#›

Computing the Median (1 of 4)

Step 1: Collect the sample data.

Starting Salaries
$44,000
$52,000
$39,000
$56,000
$61,000
$46,000
$60,000

n = 7

Slide - ‹#›

Computing the Median (2 of 4)

Step 2: Sort data from smallest to largest.

Starting Salaries
$39,000
$44,000
$46,000
$52,000
$56,000
$60,000
$61,000

n = 7

Slide - ‹#›

Computing the Median (3 of 4)

Step 3: Calculate the median index.

If i is not an integer, round up to next highest integer.

If i is an integer, the median is the average of the values in position i and i + 1.

The Median is the 4th value from top or bottom of the data.

Slide - ‹#›

Computing the Median (4 of 4)

Step 4: Find the median.

Slide - ‹#›

Median Example - Even Number of Data

20	5	10	40	50
15	30	20	20	90

Sales Data

Number of Sales Made n = 10

Sort the Data

When there are an even number of data points, the median is the average of the middle two values.

Slide - ‹#›

Median - Impact of Extreme Values Sales Data Example

Extreme

Value

The Median is Unaffected by Extreme Values.

Slide - ‹#›

Median - Impact of Extreme Values Accounting Graduates Starting Salaries

The only reason that the two medians are not equal is that the number of data values is different – Extreme salary did not affect the median.

Slide - ‹#›

Comparing the Mean and the Median

When extreme values are present, the mean gets pulled above the median.

Slide - ‹#›

Characteristics of the Median

Not Sensitive to Extreme Values

Uses only the Middle Value(s)

Slide - ‹#›

Mode (1 of 2)

Mode: The value in the data which appears most frequently

Sales Data

20	5	10	40	50
15	30	20	20	90

Mode = 20 (occurs 3 times)

Slide - ‹#›

The Mode

Number of Students Absent from Class Per Day

Sample Data

The Mode can be a weak measure of the center.

Slide - ‹#›

Mode: Example N B A Player Weights

Slide - ‹#›

Mode: Example N B A Player Weights Using Excel 2016

Slide - ‹#›

Mode (2 of 2)

The value in a data set that occurs most frequently

Is not affected by extreme values.

Can be used for both quantitative and qualitative data (ratio, interval, ordinal, nominal.)

Can have more than one mode, or no mode.

Distribution with two modes - bimodal

Slide - ‹#›

Distribution Shapes Relationship to Mean, Median and Mode (1 of 2)

Symmetrical Distribution

Slide - ‹#›

Distribution Shapes Relationship to Mean, Median and Mode (2 of 2)

Slide - ‹#›

Weighted Mean

The mean value of data values that have been weighted according to their relative importance

Weighted Mean for a Population

Weighted Mean for a Sample

- The weight of the ith data value

- The ith data value

Slide - ‹#›

Weighted Mean Example: Financial Portfolio

Mutual Fund	Percentage Retrun	Shares
Blank	x	w
Fidelity	7.2	2,000
Vanguard	8.3	5,000
Dimensional	5.4	12,000

Slide - ‹#›

Percentiles and Quartiles

Percentiles

The pth percentile in a data array:

p% are less than or equal to this value.

(100 − p)% are greater than or equal to this value.

50th percentile is the median.

Quartiles

1st quartile = 25th percentile

2nd quartile = 50th percentile

Also the median

3rd quartile = 75th percentile

Slide - ‹#›

Calculating Percentiles

Step 1: Sort the data in order from the lowest to highest value.

Step 2: Determine the percentile location index:

Step 3: If i is not an integer, then round to next highest integer. The pth percentile is located at the rounded index position. If i is an integer, the pth percentile is the average of the values at location index positions i and i + 1.

Slide - ‹#›

Calculating Percentiles Example (1 of 2)

Moving Company Travel Distances - Miles

13.5	8.6	16.2	21.4	21.0	23.7	4.1	13.8	20.5	9.6
11.5	6.5	5.8	10.1	11.1	4.4	12.2	13.0	15.7	13.2
13.4	13.1	21.7	14.6	14.1	12.4	24.9	19.3	26.9	11.7

Determine the 80th Percentile.

Slide - ‹#›

Calculating Percentiles Example (2 of 2)

Step 1 Sort the data from lowest to highest.

Step 2 Determine percentile location index, i, using Equation 3.6. The 80th percentile location index is

80th percentile is the average of the 24th and 25th values.

Slide - ‹#›

Calculating Percentiles Using Excel 2016 - NBA Player Weights Example

Slide - ‹#›

Calculating Quartiles

Find the 1st quartile in an ordered array of 19 values.

Quartile location index:

Use value at 5th position.

1st quartile

equals 51.

Slide - ‹#›

Calculating Quartiles Using Excel 2016 – N B A Player Weights Example

Slide - ‹#›

Box and Whisker Plot

A graph that is composed of two parts: a box and the whiskers

The box has a width that ranges from the first quartile

to the third quartile

A vertical line through the box is placed at the median.

Limits are located at a value that is 1.5 multiplied by the

difference between

The whiskers extend to the left to the lowest value within the limits and to the right to the highest value within the limits.

Slide - ‹#›

Constructing a Box and Whisker Plot (1 of 2)

Step 1: Sort values from lowest to highest.

Step 2: Find

Step 3: Draw the box so that the ends are at

Step 4: Draw a vertical line through the median.

Step 5: Calculate the interquartile range

Step 6: Extend dashed lines from each end to the highest and lowest values within the limits.

Step 7: Identify outliers with an asterisk (*).

Slide - ‹#›

Constructing a Box and Whisker Plot (2 of 2)

The center box extends from

The line within the box is the median.

The whiskers extend to the smallest and largest values within the calculated limits.

Outliers are plotted outside the calculated limits.

Slide - ‹#›

Box and Whisker Plot – Example Miles Between Fill-Ups

231	236	241	242	242	243	243	243	248
248	249	250	251	251	252	252	254	255
255	256	256	257	259	260	260	260	260
262	262	264	265	265	265	266	268	268
270	276	277	277	280	286	300	324	345

Slide - ‹#›

Box and Whisker Plot – Example A T M Usage by Location

Slide - ‹#›

Data Level Issues

Be aware of the level of data before computing a numerical measure.

Slide - ‹#›

The Mean - Data Level Requirements

Ratio or Interval Level Data

Variables Such As:

Age

Income

Interest Rate

Stock Prices

Sales

Number of Defects

Temperature

etc.

Slide - ‹#›

The Mean – Caution with Ordinal Data (1 of 2)

Computing a Mean for Ordinal Data – Not Recommended

Example: Ordinal Data − Education Level

1 = Grade School

2 = High School

3 = Some College

4 = College Degree

5 = Graduate Degree

Sample of n = 5 people − Education Codes

Does this imply that the mean education level in the sample is somewhere between High School and Some College?

Slide - ‹#›

The Mean – Caution with Ordinal Data (2 of 2)

Example: 5 Point Scale Question

I believe that I spend enough time studying for my statistics class.

Responses

2
4
3
5
2
1
3
3
4

Concerns:

Is there an equal distance between the 5 categories – is the distance in meaning between S A and A the same as between A and Neutral?

Is a response of A twice as high as a response of D?

The mean computation assumes the answer to both questions is “Yes”.

Slide - ‹#›

The Mean – Definitely Not with Nominal Data

What is your Marital Status?

1 = Single

2 = Married

3 = Divorced

4 = Widowed

Mean = 1.93 Does this mean that on average, a person in the sample is almost married??

Don’t compute the mean for ordinal data!

Slide - ‹#›

The Median - Data Level Requirements

Ratio or Interval Level Data

Variables Such As:

Age

Income

Interest Rate

Stock Prices

etc.

Okay for Ordinal Data

Not for Nominal Data

Slide - ‹#›

Descriptive Measures - Summary

Descriptive Measure	Computation Method	Data Level	Advantages/Disadvantages
Mean	Sum of values divided by the number of values	Ratio Interval	Numerical center of the data Sum of deviations from the mean is zero Sensitive to extreme values
Median	Middle value for data that have been sorted	Ratio Interval Ordinal	Not sensitive to extreme values Computed only from the center values Does not use information from all the data
Mode	Value(s) that occur most frequently in the data	Ratio Interval Ordinal Nominal	May not reflect the center May not exist Might have multiple modes

Slide - ‹#›

Comparing Distributions Production Volumes Example

Slide - ‹#›

Production Distributions What Is the Difference?

Slide - ‹#›

Section 3.2 Measures of Dispersion/Variation

Slide - ‹#›

Descriptive Statistics Measures of Dispersion

Range

Simple range

Interquartile Range

Variance

Standard Deviation

Coefficient of Variation

Slide - ‹#›

Range

A measure of variation that is computed by finding the difference between the maximum and minimum values in a data set

R = Maximum Value − Minimum Value

Simplest measure of variation

Is very sensitive to extreme values

Ignores the data distribution

Slide - ‹#›

Measures of Dispersion - Staff Years of Experience

Raw Data

Values are the number of years experience for a sample of staff members.

13	16	16	13	13	9
12	15	8	12	13	14
11	6	15	11	15	13
12	12	13	13	13	11
17	14	11	14	13	15
11	11	13	15	9	10
11	12	11	16	11	10
13	13	11	13	10	13
11	10	8	11	11	17
10	12	13	10	10	15
14	11	9	15	13	12
11	14	12	10	12	14
14	12	11	11	12	13
11	11	11	13	12	11
12	10	9	12	10	13
12	14	10	10	16	12
11	11	14	14	Blank	Blank

n = 100

Slide - ‹#›

Finding the Range

Years Experience

Range = High − Low

Range = 17 − 6 = 11

Slide - ‹#›

Characteristics of the Range

Easy to Compute

Uses Only Two Values (high and low)

Sensitive to Extremes

Slide - ‹#›

Interquartile Range

A measure of variation that is determined by computing the difference between the third and first quartiles

Interquartile Range

Eliminates outlier problems

Eliminates some high- and low-valued observations

Slide - ‹#›

Interquartile Range Example

Interquartile range = 57 − 30 = 27

Slide - ‹#›

Population Variance

The average of the squared distances of the data values from the mean.

- population mean, N – population size

Slide - ‹#›

Population Variance - Example

Slide - ‹#›

Measures of Dispersion Production Volume Example

Departments the same in terms of the center but differ in degree of dispersion

Slide - ‹#›

Computing the Variance Department A

Slide - ‹#›

Computing the Variance Department B

Slide - ‹#›

Comparing the Dispersion Dept. A vs Dept. B

The variance is measured in units squared.

Slide - ‹#›

Population Standard Deviation

The most commonly used measure of variation

The positive square root of the variance

Has the same units as the original data

Slide - ‹#›

Standard Deviation (The Square Root of the Variance)

Population Standard Deviation

Dept. A

Dept. B

Standard deviation is the measure of average deviation from the mean.

Slide - ‹#›

Population Standard Deviation – Example Using Excel 2016

The Convention Center hosted 20 events last year. This is the population.

The mean attendance is 381.55

The standard deviation is 269.00

Slide - ‹#›

Sample Standard Deviation

sample variance

sample standard deviation

Slide - ‹#›

Sample Standard Deviation Restaurant Sales Example

Restaurant Sales Values (n = 7)

$30

$19.50

$22.40

$27

$17.50

$25.40

$23.50

s = $4.31

Slide - ‹#›

Sample Standard Deviation – Example Using Excel 2016 (1 of 2)

Sample of starting salaries for college graduates

n = 7

Slide - ‹#›

Sample Standard Deviation – Example Using Excel 2016 (2 of 2)

Fitness Club Membership Ages

Slide - ‹#›

Interpreting the Standard Deviation Health Club Example

Standard deviation is a measure of distance from the mean.

Slide - ‹#›

Section 3.3 Using the Mean and Standard Deviation Together

Coefficient of Variation (C V)

The ratio of the standard deviation to the mean expressed as a percentage. The coefficient of variation is used to measure variation relative to the mean.

Measures relative variation

Always expressed in percentage (%)

Shows variation relative to mean

Slide - ‹#›

Coefficient of Variation

Is used to compare two or more sets of data measured in different units

Population C V

Sample C V

Slide - ‹#›

Comparing Coefficients of Variation

Stock A:

Average Price = $50 =

Standard Deviation = $5 = s

Stock B:

Average Price = $100 =

Standard Deviation = $5 = s

Both stocks have the same standard deviation, but stock B is less variable relative to its price.

Slide - ‹#›

The Empirical Rule (1 of 2)

If the data distribution is bell shaped, then the interval

contains approximately 68% of the values

contains approximately 95% of the values

contains virtually all of the data values.

Slide - ‹#›

The Empirical Rule (2 of 2)

Slide - ‹#›

The Empirical Rule - Example

The travel distances for the Uber service in a major city is a bell shaped distributed with a mean of 15.1 miles and standard deviation of 3.1 miles per trip. Use the Empirical Rule to describe the distribution.

Assuming a bell shaped distribution, 68% percent of the trips should be between:

Assuming a bell shaped distribution, 95% percent of the trips should be between:

Slide - ‹#›

Tchebysheff’s Theorem

Regardless of how data are distributed, at least

of the values will fall within k standard

deviations of the mean.

Examples:

Slide - ‹#›

Standardized Data Values

The number of standard deviations a value is from the mean

Standardized data values are also referred to as z scores.

Population z score

Sample z score

x – data value

μ – population mean

σ – population standard deviation

– sample mean

s – sample standard deviation

Slide - ‹#›

Converting Data to Standardized Values

Step 1: Collect the population or sample values for the quantitative variable of interest.

Step 2: Compute the population mean and standard deviation or the sample mean and standard deviation.

Step 3: Convert the values to standardized z-values.

Slide - ‹#›

Standardized Value Calculation - Example

I Q Scores have a distribution thought to be bell shaped with a mean equal to 100 and standard deviation equal to 15. Suppose a person has an I Q of 121, calculate the standardized I Q.

A person with an I Q of 121 has an I Q that is 1.40 standard deviations higher than the population mean I Q.

Slide - ‹#›

Data Standardization – Example University Entrance Test

How did you do on a relative basis at each university?

Slide - ‹#›

Entrance Test Scores

Entrance Score Was Relatively Better at University A than at University B

Slide - ‹#›

University Entrance Score Analysis

Slide - ‹#›

Coefficient of Variation (C V) – Example University Entrance Scores

The Coefficient of Variation measures the relative dispersion in a set of data. The C V is used to compare two or more sets of data with respect to variation.

Population

University A

University B

University A had more variable entrance scores.

Slide - ‹#›

216255330.....310295

++++

2,884

288.4

2213101623131113

+++++++

121

15.125

20510405090

+++++

300

30.0

300

===

(

)

73.5

(

)

2020

Median 20

(10)5

(

)

2020

Median 20

(

)

2020

Median 20

1 1 1 1 1 1 2 2 3 3 3 4 4 5

(7.2)(2,000)(8.3)(5,000)(5.4)(12,000)

6.35%

2,0005,00012,000

===

(0100)

££

where

()

100

()(30)24

100100

===

20.521

20.75

36404246515662657174788284878890929597

()(19)4.75

100100

===

()

().

1313

QQQQ

andbelowandabove

123

,,.

QQQ

and

(

)

IQRQQ

14342

2.8????

===

3????

===

()

(

)

===

(

)

400

===

()

(

)

.40.632

808.94

( )

()

165.3

23.61

===

(

)

(

)

(

)

222

()

3023.6119.523.61....23.523.61

171

-+-+-

11.18

34.84

(100)%

(100)(100)10%

===

(100)(100)5%

100

===

15.1 miles; 3.1 miles

(1)() 15.1(1)(3.1) 12.0 miles --

------- 18.2 miles

±±

(1)() 15.1(2)(3.1) 8.9 miles ---

------ 21.3 miles

±±

æö

ç÷

èø

(

)

(

)

(

)

10%...........11

175%..........22

189%.......33

æö

-==±

ç÷

èø

æö

-==±

ç÷

èø

æö

-==±

ç÷

èø

atleast

100 15

121100

1.40

===

(100)

14.2%

750

(100)

750

9.3%

231	236	241	242	242	243	243	243	248
248	249	250	251	251	252	252	254	255
255	256	256	257	259	260	260	260	260
262	262	264	265	265	265	266	268	268
270	276	277	277	280	286	300	324	345

231	236	241	242	242	243	243	243	248
248	249	250	251	251	252	252	254	255
255	256	256	257	259	260	260	260	260
262	262	264	265	265	265	266	268	268
270	276	277	277	280	286	300	324	345

231	236	241	242	242	243	243	243	248
248	249	250	251	251	252	252	254	255
255	256	256	257	259	260	260	260	260
262	262	264	265	265	265	266	268	268
270	276	277	277	280	286	300	324	345