Statistics · 2016. 7. 9. · Sample vs. Population . Sample Standard Deviation 2 ( 1) xx N ¦ V...

Statistics

Statistics

The collection, evaluation, and interpretation

of data.

Statistics

Statistics

Descriptive Statistics

Describe collected data.

Inferential Statistics

Generalize and

evaluate a population

based on sample

data.

Data

Values that possess names or labels.

(color of M&Ms, breed of dog, etc…)

Categorical or Qualitative Data

Values that represent a measurable quantity.

Population, number of M&Ms, number

of defective parts, etc.

Numerical or Quantitative Data

Data Collection

Sampling

Random

Systematic

Stratified

Cluster

Convenience

Graphic Data Representation

Histogram

Frequency Polygons

Bar Chart

Pie Chart

Frequency distribution graph

Frequency distribution graph

Categorical data graph

Categorical data graph %

Measures of Central Tendency

xx

n

Most frequently used measure of central

tendency.

Strongly influenced by outliers – very large

or very small values.

Mean

Arithmetic average.

Sum of all data values divided by the

number of data values within the array.

x


xx

n

48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55

Determine the mean value of

(48 63 62 49 58 2 63 5 60 59 55)

11x

524

11x

47.64x


Average (mean) height of NBA player:

Mean

Fun Facts:

x


Average (mean) height of NBA player:

6 feet, 7.62 inches

Mean

Fun Facts:

x


Average (mean) cost of a wedding:

Mean

Fun Facts:

x


Average (mean) cost of a wedding:

$25,200

Mean

Fun Facts:

x


Average (mean) U.S. cost of a prom:

Mean

Fun Facts:

x


Average (mean) U.S cost of a prom:

$978

Mean

Fun Facts:

x


Average (mean) life of a cell phone:

Mean

Fun Facts:

x


Average (mean) life of a cell phone:

24 months

Mean

Fun Facts:

x


Median

Data value that divides a data array into

two equal groups.

Data values must be ordered from lowest

to highest.

Useful in situations with skewed data

and outliers (e.g., wealth management).

Measures of Central Tendency Determine the median value of

Organize the data array from lowest to

highest value.

59, 60, 62, 63, 63

48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55

Select the data value that splits the data set

evenly.

2, 5, 48, 49, 55, 58,

Median = 58

What if the data array had an even number of

values? 60, 62, 63, 63 5, 48, 49, 55, 58, 59,


Median U.S. household income:

Median

Fun Facts:


Median U.S. household income:

$51,939

Median

Fun Facts:


Median home price in San Francisco:

Median

Fun Facts:


Median home price in San Francisco:

$1,000,000

Median

Fun Facts:


Usually the highest point of curve.

Mode

Most frequently occurring response within a

data array.

May not be typical.

May not exist at all.

Modal, bimodal, and multimodal.

Measures of Central Tendency Determine the mode of

48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55 Mode = 63

Determine the mode of

48, 63, 62, 59, 58, 2, 63, 5, 60, 59, 55

Mode = 63 & 59 Bimodal

Determine the mode of

48, 63, 62, 59, 48, 2, 63, 5, 60, 59, 55

Mode = 63, 59, & 48 Multimodal

Data Variation

Range

Standard Deviation

Measure of data scatter

Difference between the lowest and

highest data value.

Square root of the variance.

Range

63 2R

Calculate by subtracting the lowest value

from the highest value.

R h l

2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63

Calculate the range for the data array.

R h l

61R

s=S x-x( )

2

(n-1)

Sample Standard Deviation Population Standard Deviation.

In practice, only the sample standard deviation can be measured and therefore is more

useful for applications.

Population Standard Deviation

A population standard deviation represents a parameter, not a statistic. The standard

deviation of a population gives researchers an amount of dispersion of data for an entire

population of survey respondents.

Sample Standard Deviation

A standard deviation of a sample estimates the standard deviation of a population based

on a random sample. The sample standard deviation, unlike the population standard

deviation, is a statistic that measures the dispersion of the data around the sample

mean.

Standard Deviation – Sample vs. Population


2

( 1)

x x

N

σ for a sample, not population

1. Calculate the mean

2. Subtract the mean from each value and then

square it.

3. Sum all squared differences.

4. Divide the summation by the number of

values in the array minus 1.

5. Calculate the square root of the product.

x


2

( 1)

x x

N

2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63

Calculate the standard

deviation for the data array.

x

x

n

524

111. 47.64

2.

(2 - 47.64)2 = 2083.01

(5 - 47.64)2 = 1818.17

(48 - 47.64)2 = 0.13

(49 - 47.64)2 = 1.85

(55 - 47.64)2 = 54.17

(58 - 47.64)2 = 107.33

(59 - 47.64)2 = 129.05

(60 - 47.64)2 = 152.77

(62 - 47.64)2 = 206.21

(63 - 47.64)2 = 235.93

(63 - 47.64)2 = 235.93

2

x x


2

( 1)

x xs

N

2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63

Calculate the standard

deviation for the data array.

3.

2083.01 + 1818.17 + 0.13 + 1.85 + 54.17 + 107.33

+ 129.05 + 152.77 + 206.21 + 235.93 + 235.93

2

x x

= 5,024.55

4. 2

( 1)

x x

N

5,024.55

10 502.46

5. 2

( 1)

x xs

N

502.46

S = 22.42

2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63

Calculate the population standard

deviation for the data array

524

111. Calculate the mean

2. Subtract the mean from each data value and square each

difference


Variation 3. Sum all squared differences

2082.6777 + 1817.8595 + 0.1322 + 1.8595 + 54.2231 +

107.4050 + 129.1322 + 152.8595 + 206.3140

+ 236.0413 + 236.0413

= 5,024.5455

4. Divide the summation by the number of data values

5. Calculate the square root of the result

Note that this is the sum of the

unrounded squared differences.



Find the mean for each

group of numbers

x σ

0, 0, 14, 14

0, 6, 8, 14

6, 6, 8, 8


Find the mean for each

group of numbers

x σ

0, 0, 14, 14 7

0, 6, 8, 14 7

6, 6, 8, 8 7


Find the population standard

deviation for each group of

numbers

x σ

0, 0, 14, 14 7

0, 6, 8, 14 7

6, 6, 8, 8 7


x σ

0, 0, 14, 14 7 7

0, 6, 8, 14 7

6, 6, 8, 8 7



numbers


x σ

0, 0, 14, 14 7 7

0, 6, 8, 14 7 5

6, 6, 8, 8 7



numbers


x σ

0, 0, 14, 14 7 7

0, 6, 8, 14 7 5

6, 6, 8, 8 7 1



numbers

Graphing Frequency Distribution Numerical assignment of each outcome of a

chance experiment.

A coin is tossed 3 times. Assign the variable

X to represent the frequency of heads

occurring in each toss.

Toss Outcome X Value

HHH

HHT

HTH

THH

HTT

THT

TTH

TTT

3

2

2

2

1

1

1

0

X =1 when?

HTT,THT,TTH

Graphing Frequency Distribution

The calculated likelihood that an outcome

variable will occur within an experiment.

Toss Outcome X value

HHH

HHT

HTH

THH

HTT

THT

TTH

TTT

3

2

2

2

1

1

1

0

x P(x)

0

1

2

3

xx

a

FP

F

0

1P

8

1

3P8

2

3P

8

3

1P

8

Graphing Frequency Distribution

x P(x)

0

1

2

3

0

1P

8

1

3P8

2

3P

8

3

1P

8 x

Histogram

Histogram Available airplane passenger seats one week

before departure.

What information does

the histogram provide

the airline carriers?

What information

does the histogram

provide prospective

customers?

open seats

perc

ent of th

e tim

e

Statistics · 2016. 7. 9. · Sample vs. Population . Sample Standard Deviation 2 ( 1) xx N ¦ V...

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