Statistics

StatisticsThe collection, evaluation, and interpretation of data

StatisticsStatistics

Statistics

Descriptive Statistics

Describe collected data

Inferential Statistics

Generalize and evaluate a population based on sample data

DataData

Values that possess names or labelsColor of M&Ms, breed of dog, etc.

Categorical or Qualitative Data

Values that represent a measurable quantityPopulation, number of M&Ms, number of defective parts, etc.

Numerical or Quantitative Data

DataData CollectionCollectionSampling

Random

Systematic

Stratified

Cluster

Convenience

Graphic Data RepresentationGraphic Data RepresentationHistogram

Frequency Polygons

Bar Chart

Pie Chart

Frequency distribution graph

Frequency distribution graph

Categorical data graph

Categorical data graph %

Measures of Central TendencyMeasures 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

Measures of Central TendencyMeasures of Central Tendency

xx

n

48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55Determine the mean value of

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

11

524x

11

x 47.64

Measures of Central TendencyMeasures of Central TendencyMedian

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 TendencyMeasures of Central TendencyDetermine 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, 635, 48, 49, 55, 58, 59,

Measures of central tendencyMeasures of central tendency

Usually the highest point of curve

ModeMost frequently occurring response within a data array

May not be typical

May not exist at all

Mode, bimodal, and multimodal

Measures of Central TendencyMeasures of Central TendencyDetermine the mode of

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

Determine the mode of

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

Determine the mode of

48, 63, 62, 59, 48, 2, 63, 5, 60, 59, 55Mode = 63, 59, & 48 Multimodal

Data VariationData Variation

Range

Standard Deviation

Variance

Measure of data scatter

Difference between the lowest and highest data value

Square root of the variance

Average of squared differences between each data value and the mean

RangeRange

R 63 2

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

R 61

Standard DeviationStandard Deviation 2x xs

( N 1)

1. Calculate the mean .

2. Subtract the mean from each value.

3. Square each difference.

4. Sum all squared differences.

5. Divide the summation by the number of values in the array minus 1.

6. Calculate the square root of the product.

x


( N 1)

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

Calculate the standard deviation for the data array.

x

x

n

52411

1. 47.64

2. 2 - 47.64 = -45.64

5 - 47.64 = -42.64

48 - 47.64 = 0.36

49 - 47.64 = 1.36

55 - 47.64 = 7.36

58 - 47.64 = 10.36

59 - 47.64 = 11.36

60 - 47.64 = 12.36

62 - 47.64 = 14.36

63 - 47.64 = 15.36

63 - 47.64 = 15.36

x x


( N 1)

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


3.

-45.642 = 2083.01

-42.642 = 1818.17

0.362 = 0.13

1.362 = 1.85

7.362 = 54.17

10.362 = 107.33

11.362 = 129.05

12.362 = 152.77

14.362 = 206.21

15.362 = 235.93

15.362 = 235.93

2x x


( N 1)

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


4.

2083.01 + 1818.17 + 0.13 + 1.85 + 54.17 + 107.33 + 129.05 + 152.77 + 206.21 + 235.93 + 235.93

2x x

= 5,024.555.( N 1)

11-1 = 10

6. 2( 1

x x

N )

5,024.5510

502.46

7. 2x xs

( N 1)

502.46S = 22.42

VarianceVariance 22x x

s( N 1)

1.Calculate the mean.

2.Subtract the mean from each value.

3.Square each difference.

4.Sum all squared differences.

5.Divide the summation by the number of values in the array minus 1.

Average of the square of the deviations

VarianceVariance

2 5024.55s

(50

1 )46

02.

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

Calculate the variance for the data array.

22x x

s( N 1)

Graphing Frequency DistributionGraphing Frequency DistributionNumerical 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 DistributionGraphing Frequency DistributionThe 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

3P

8

2

3P

8

3

1P

8

Graphing Frequency DistributionGraphing Frequency Distribution

x P(x)

0

1

2

3

0

1P

8

1

3P

8

2

3P

8

3

1P

8 x

HistogramHistogram

HistogramHistogramOpen airplane passenger seats one week before departure

What information does the histogram provide the airline carriers?

What information does the histogram provide prospective customers?

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