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DESCRIPTION
Formulae for calculating central tendencies
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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
Standard DeviationStandard Deviation 2x xs
( 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
Standard DeviationStandard Deviation 2x xs
( N 1)
2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63
Calculate the standard deviation for the data array.
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
Standard DeviationStandard Deviation 2x xs
( N 1)
2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63
Calculate the standard deviation for the data array.
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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