1. Statistics
Principles of Engineering
© 2012 Project Lead The Way, Inc.

2. Principles of Engineering – Statistics
As a class, we will look at this clip: Get your Notebook. In your notebook, draw the following slides (use colors) and make sure they are 4 per page: , 5, 6, 7 8, 9, 10, , 13, 14 and 15

3. Statistics The collection, evaluation, and interpretation of data
Principles of EngineeringTM
Unit 4 – Lesson Statistics
The collection, evaluation, and interpretation of data

4. Statistics Statistics Descriptive Statistics Inferential Statistics
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Statistics
Descriptive Statistics
Describe collected data
Inferential Statistics
Generalize and evaluate a population based on sample data
Predictions are a type of inferential statistics.

5. 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&M’s, breed of dog, etc.
Numerical or Quantitative Data
Values that represent a measurable quantity
Population, number of M&M’s, number of defective parts, etc.

6. 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, random sampling is practiced when you put each student’s name in a hat and draw one at random.
Systematic sampling involve selecting individuals at regular intervals. For instance, choose every fourth name on the roll sheet for your class.
Stratified sampling ensures that you equally represent certain subgroups. For instance, you may randomly choose two males and two 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, if you ask a few of your friends for feedback, you are performing a convenience sample.

7. 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 %

8. Measures of Central Tendency
Statistics
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

9. 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

10. 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)

11. Measures of Central Tendency
Statistics
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

12. Measures of Central Tendency
Statistics
Measures of Central Tendency
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
Modal, bimodal, and multimodal

13. 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

14. Data Variation Range Standard Deviation Measure of data scatter
Statistics
Data Variation
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

15. 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

16. Standard Deviation – Sample vs. Population
Statistics
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Sample Standard Deviation Population Standard Deviation.
In practice, only the sample standard deviation can be measured and therefore is more useful for applications.
σ= x i − μ 2 N
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.
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.
Levy Sarfin, R. (n.d.). What is the difference between sample & population standard deviation? Retrieved September 23, 2014.
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.

17. Sample Standard Deviation
Statistics
Principles of EngineeringTM
Unit 4 – Lesson Statistics
s for a sample, not population
Calculate the mean
Subtract the mean from each value and then square it.
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.

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

19. Sample Standard Deviation
Statistics
Sample 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 6. 5.
s = 22.42

20. Population Standard Deviation
Introduction to Summary Statistics
σ= x i − μ 2 N
Calculate the population standard deviation for the data array
2, 5, 48, 49, 55, 58, 59, 60, 62, 63, 63
μ = x i N
=47.64
1. Calculate the mean
2. Subtract the mean from each data value and square each difference
x i − μ 2
( )2 =
( )2 =
( )2 =
( )2 =
( )2 =
( )2 =
( )2 =
( )2 =
( )2 =
( )2 =
Since we are given a finite data set and are not told otherwise, we assume we have a data value for each member of the entire population. We use the POPULATION standard deviation.
Find the mean of the data. [3 clicks] NOTE that if we were asked for the mean, we would report the mean to be However, we will use the unrounded value stored in the calculator when calculating the standard deviation.
Find the difference between the mean and each data value and square each difference. [many clicks] Note that these values are calculated using the unrounded mean, but they are rounded to two decimal places since it is unwieldy to report to a large number of decimal places. It is preferable to save each UNROUNDED squared difference in your calculator so that we can add them together.

21. Introduction to Summary Statistics
Population Standard Deviation
Introduction to Summary Statistics
Variation
3. Sum all squared differences
x i − μ 2 =
Note that this is the sum of the unrounded squared differences.
= 5,
4. Divide the summation by the number of data values
x i − μ 2 N = =
4. Sum the squared differences. [click] Note that the sum shown is calculated using the unrounded values (and so the total is slightly different than the total you get when adding the rounded numbers). If you use the rounded squared differences, the sum is nearly the same
5. Divide the sum by the number of data values. [click] Remember to use the unrounded result from the previous step (even though it does not matter in this case).
6. Take the square root of the division. [click] Remember to use the unrounded result from the previous step.
Report the standard deviation to one more digit than the original data.
5. Calculate the square root of the result
x i − μ 2 N = = 21.4

22. Graphing Frequency Distribution
Statistics
Graphing Frequency Distribution
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Numerical assignment of each outcome of a chance experiment
A coin is tossed three 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

23. 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 Px
HHH 3
HHT 2
HTH 2 1
THH 2
HTT 1 2
THT 1
TTH 1 3
TTT

24. Graphing Frequency Distribution
Statistics
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Histogram x Px 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

25. Histogram Available airplane passenger seats one week before departure
Statistics
Histogram
Principles of EngineeringTM
Unit 4 – Lesson Statistics
Available airplane passenger seats one week before departure
What information does the histogram provide the airline carriers?
percent of the time
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.
Note that something is wrong with this relative frequency graph! The bars we see only add to a total of about 0.50 (50 percent), and they should add to 100 percent.
open seats