Statistics: Understanding Populations, Samples, and Descriptive Statistics, Slides of Engineering Mathematics

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Statistics

Dealing With Uncertainty

Objectives

• Describe the difference between a sample and a population
• Learn to use descriptive statistics (data sorting, central tendency, etc.)
• Learn how to prepare and interpret histograms
• State what is meant by normal distribution and standard normal distribution.
• Use Z-tables to compute probability.

Statistics is...

• a standard method for...
  • collecting, organizing, summarizing, presenting, and analyzing data
  • drawing conclusions
  • making decisions based upon the analyses of these data.
• used extensively by engineers (e.g., quality control)

Populations and Samples

• Population - complete set of all of the possible instances of a particular object - e.g., the entire class
• Sample - subset of the population
  • e.g., a team
• We use samples to draw conclusions about the parent population.

Team Exercise: Sample Bias

• To three significant figures, estimate the average age of the class based upon your team.
• When would a team not be a representative sample of the class?

Measures of Central Tendency

• If you wish to describe a population (or a sample) with a single number, what do you use? - Mean - the arithmetic average - Mode - most likely (most common) value. - Median - “middle” of the data set.

Sample Mean

Where:

• is the sample mean
• xi are the data points
• n is the sample size



n

i

x (^) n xi 1

x

Population Mean

Where:

• μ is the population mean
• x i are the data points
• N is the total number of observations in the population

 



N i N xi 1

^1

Mode continued

• Example of a grade distribution with mean C, mode B

0

5

10

15

20

25

F D C B A

What is the Median?

• Median - for sorted data, the median is the middle value (for an odd number of points) or the average of the two middle values (for an even number of points). - useful to characterize data sets with a few extreme values that would distort the mean (e.g., house price,family incomes).

Standard Deviation

• Gives a unique and unbiased estimate of

the scatter in the data.

Standard Deviation

• Population
• Sample

2 1

 (^1) (  ) 

 

N i^ i

x N

2 1

( ) ( 1 )

(^1) x x n

s

n i

 i 

 



Deviation

Variance = ^2

Variance = s^2

A Valuable Tool

• Gauss invented standard deviation circa 1700 to explain the error observed in measured star positions.
• Today it is used in everything from quality control to measuring financial risk.

Team Exercise

• In your team’s bag of M&M candies, count
  • the number of candies for each color
  • the total number of candies in the bag
• When you are done counting, have a representative from your team enter your data on the board
• Using Excel, enter the data gathered by the entire class More