1. Statistics Statistics- Inferential Statistics Descriptive Statistics
Principles of EngineeringTM
Unit 4 – Lesson Statistics
The collection, evaluation, and interpretation of data
Statistics
Inferential Statistics
Generalize and evaluate a population based on sample data
Descriptive Statistics
Describe collected data
Predictions are a type of inferential statistics.

2. Data Categorical or Qualitative Data
Statistics
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Categorical or Qualitative Data
Values that possess names or labels Color of M&Ms, breed of dog, etc.
Numerical or Quantitative Data
Values that represent a measurable quantity
Population, number of M&Ms, number of defective parts, etc.

3. Data Collection Sampling Random Systematic Stratified Cluster
Statistics
Data Collection
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Sampling
Random
Systematic
Stratified
Cluster
Convenience
Random sampling involves choosing individuals completely at random from a population- for instance putting each student’s name in a hat and drawing one at random.
Systematic Sampling involve selecting individuals at regular intervals. For instance, choose every 4th name on the roll sheet for your class.
Stratified sampling makes sure you’re equally representing certain subgroups: for instance, randomly choose 2 males and 2 females in your class
Cluster sampling involves picking a few areas and sampling everyone in those areas. For instance, sample everyone in the first row and everyone in the third row, but no one else.
A convenience sample follows none of these rules in particular: for instance, ask a few of your friends.

4. Graphic Data Representation
Statistics
Graphic Data Representation
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Histogram
Frequency distribution graph
Frequency Polygons
Frequency distribution graph
Bar Chart
Categorical data graph
Histograms, bar charts, and pie charts are generally used for categorical data.
Frequency polygons are often used for numerical data
Pie Chart
Categorical data graph %

5. Measures of Central Tendency
Statistics
Measures of Central Tendency
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Mean
Arithmetic average
Sum of all data values divided by the number of data values within the array
Most frequently used measure of central tendency
Strongly influenced by outliers- very large or very small values

6. Measures of Central Tendency
Statistics
Measures of Central Tendency
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Determine the mean value of
48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55

7. Measures of Central Tendency
Statistics
Principles of EngineeringTM
Unit 4 – Lesson Statistics
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)

8. Measures of Central Tendency
Statistics
Measures of Central Tendency
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Determine the median value of
48, 63, 62, 49, 58, 2, 63, 5, 60, 59, 55
Organize the data array from lowest to highest value.
2, 5, 48, 49, 55,
58,
59, 60, 62, 63, 63
Select the data value that splits the data set evenly.
If the data array has an even number of values, we take the average (mean) of the two middlemost values. In the example, this is 58.5
Median = 58
What if the data array had an even number of values?
5, 48, 49, 55,
58, 59,
60, 62, 63, 63

9. Measures of central tendency
Statistics
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Mode
Most frequently occurring response within a data array
Usually the highest point of curve
May not be typical
May not exist at all
Mode, bimodal, and multimodal

10. Measures of Central Tendency
Statistics
Measures of Central Tendency
Principles of EngineeringTM
Unit 4 – Lesson Statistics
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, & Multimodal

11. Data Variation Range Standard Deviation Variance
Statistics
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Measure of data scatter
Range
Difference between the lowest and highest data value
Standard Deviation
Square root of the variance
Variance
Average of squared differences between each data value and the mean

12. Statistics
Range
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Calculate by subtracting the lowest value from the highest value.
Calculate the range for the data array.
2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63

13. Standard Deviation Calculate the mean .
Statistics
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Calculate the mean .
Subtract the mean from each value.
Square each difference.
Sum all squared differences.
Divide the summation by the number of values in the array minus 1.
Calculate the square root of the product.
Note that this is the formula for the “sample standard deviation”, which statisticians distinguish from the “population standard deviation”. In practice, only the sample standard deviation can be measured, and therefore is more useful for applications.

14. Standard Deviation
Statistics
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Calculate the standard deviation for the data array.
2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63 1. 2. = = = = = = = = = =

15. Standard Deviation
Statistics
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Calculate the standard deviation for the data array.
2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63 3. = =
0.362 =
1.362 =
7.362 = = = = = =

16. Statistics
Standard Deviation
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Calculate the standard deviation for the data array.
2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63 4.
= 5,024.55 7. 5.
11-1 = 10 6.
S = 22.42

17. Variance Average of the square of the deviations Calculate the mean.
Statistics
Variance
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Average of the square of the deviations
Calculate the mean.
Subtract the mean from each value.
Square each difference.
Sum all squared differences.
Divide the summation by the number of values in the array minus 1.
This is the sample variance (the square of the sample standard deviation). Note that we don’t need this formula- we just found S, and the variance is S^2, so we can find this directly.

18. Variance Calculate the variance for the data array.
Statistics
Variance
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Calculate the variance for the data array.
2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63

19. Graphing Frequency Distribution
Statistics
Principles of EngineeringTM
Unit 4 – Lesson Statistics
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 3
X =1 when?
HHT 2
HTH 2
HTT,THT,TTH
THH 2
HTT 1
THT 1
TTH 1
TTT

20. Graphing Frequency Distribution
Statistics
Principles of EngineeringTM
Unit 4 – Lesson Statistics
The calculated likelihood that an outcome variable will occur within an experiment
Toss Outcome
X value x
P(x)
HHH 3
HHT 2
HTH 2 1
THH 2
HTT 1 2
THT 1
TTH 1 3
TTT

21. Graphing Frequency Distribution
Statistics
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Histogram x
P(x) 1 2
Because this is a frequency histogram, the heights of all the bars must add to 1 (1/8 + 3/8 + 3/8 + 1/8 = 1). x 3

22. Histogram Open airplane passenger seats one week before departure
Statistics
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Open airplane passenger seats one week before departure
What information does the histogram provide the airline carriers?
What information does the histogram provide prospective customers?
Airline carriers and passengers can see how many seats will likely be open on a flight one week prior to departure.
For instance (looking at the tallest bar) 12 percent of the time there are 5 empty seats.
For some reason the graph does not show the likelihood of zero empty seats, but it is probably quite high, since the bars we see only add to a total of about .50 (50 percent).