1. Probability & Statistics
Probability & Statistics
Week 01
Introduction to Statistics

2. What is Statistic?

3. Statistics Statistics is the science of data which involves
1-2
Statistics
Statistics is the science of data which involves
collecting,
classifying,
summarizing,
organizing,
analyzing,
and interpreting numerical information

5. Data collection methods
1-2
Data collection methods
Questionnaires.
Interviews
Observation

6. Why study Statistic? You’ll be able to make objective decisions,
You’ll be able to
make objective decisions,
make accurate predictions that seem inspired
convey the message you want in the most effective way possible.
Statistics can be a convenient way of summarizing key truths about data
need a way of visualizing data for everyone else.

7. Why not just go on the data? Why chart it?
Why not just go on the data? Why chart it?
Sometimes it’s difficult to see what’s really going on just by looking at the raw data.
There can be patterns and trends in the data, but these can be very hard to spot if you’re just looking at a heap of numbers.
Charts give you a way of literally seeing patterns in your data.
They allow you to visualize your data and see what’s really going on in a quick glance.

10. What’s the difference between information and data?
What’s the difference between information and data?
Data refers to raw facts and figures that have been collected.
Information is data that has some sort of added meaning.

11. Definitions Populations and Parameters
Definitions
Populations and Parameters
A population is the entire collection of all observations of interest.
E.g. All 2.5 million registered voters in Sri Lanka
A parameter is a descriptive measure of the entire population of all observations of interest

12. Definitions Samples and Statistics
Definitions
Samples and Statistics
A sample is a representative portion of the population which is selected for study.
Potentially very large, but less than the population.
E.g. a sample of 765 voters exit polled on election day.
A statistic describes a sample and serves as an estimate of the corresponding population parameter.

14. relationship between samples and populations.

15. Parameters are numbers that summarize data for an entire population.
Parameters are numbers that summarize data for an entire population.
Statistics are numbers that summarize data from a sample, i.e. some subset of the entire population
Eg:
A nutritionist wants to estimate the mean amount of sodium consumed by children under the age of 10. From a random sample of 75 children under the age of 10, the nutritionist obtains a sample mean of 2993 milligrams of sodium consumed.

16. Individuals and Variables
Individuals and Variables
Individuals are the people or objects included in the study.
A variable is the characteristic of the individual to be measured or observed.
For example,
if we want to do a study about the people who have
climbed Mt. Everest, then the individuals in the study are the actual people who made it to the top. The variables to measure or observe might be the height, weight, race, gender, income, etc of the individuals that made it to the top of Mt. Everest.

17. Definitions Variables
Definitions
Variables
A variable is a the characteristic of the population that is being examined in the statistical study.
There are two basic types of data: Qualitative & Quantitative

18. Types of Variables
Qualitative or Attribute variable(Categorical): the characteristic or variable being studied is nonnumeric.
EXAMPLES: Gender, religious affiliation, type of automobile owned, state of birth, eye color, type of dessert

19. Types of Variables
Quantitative variable: the variable can be reported numerically.
EXAMPLE: balance in your savings account, minutes remaining in class, number of children in a family.

20. Types of Variables
Quantitative variables can be classified as either discrete or continuous.
Discrete variables: can only assume certain values and there are usually “gaps” between values.
EXAMPLE: the number of bedrooms in a house. (1,2,3,..., etc...).
Continuous variables: can assume any value within a specific range.
EXAMPLE: The time it takes to fly from Sri Lanka to New York.

21. Types of Statistics Descriptive Statistics:
Types of Statistics
Descriptive Statistics:
Methods of organizing, summarizing, and presenting data in an informative way.
Descriptive statistics do not allow us to make conclusions beyond the data we have analyzed or reach conclusions regarding any hypotheses we might have made.
Frequency distributions, measures of central tendency (mean, median, and mode), and graphs like pie charts and bar charts that describe the data are all examples of descriptive statistics.

22. EXAMPLE for Descriptive Statistics:
EXAMPLE for Descriptive Statistics:
if we look at a basketball team's game scores over a year, we can calculate the average score, variance etc. and get a description (a statistical profile) for that team
According to Consumer Report of Ceylon Pencil Company, 9 defective pens per 100. The statistic 9 describes the number of problems out of every 100 pens

23. Types of Statistics Inferential Statistics:
Types of Statistics
Inferential Statistics:
Inferential statistics is concerned with making predictions or inferences about a population from observations and analyses of a sample
The methods of inferential statistics are (1) the estimation of parameter(s) and (2) testing of statistical hypotheses. A Chi-square or T-test

24. Inferential Statistics:
Inferential Statistics:
EXAMPLE:
TV networks constantly monitor the popularity of their programs by hiring people to sample the preferences of TV viewers.
To infer the success rate of a drug in treating high temperature, by taking a sample of patients, giving them the drug, and estimating the rate of effectiveness in the population using the rate of effectiveness in the sample.

25. 1-14
Levels of Measurement
There are four levels of measurement: nominal, ordinal, interval and ratio.

26. 1-13
Nominal level
Nominal level (scaled): Data that can only be classified into categories and cannot be arranged in an ordering scheme.
EXAMPLES: eye color, gender, religious affiliation
Religion (Catholic, Buddhist, etc)
Race ( African-American, Asian, etc)
Marital Status (Married, Single, Divorced)
These categories are mutually exclusive and/or exhaustive.

27. Nominal level
Mutually exclusive: An individual or item that, by virtue of being included in one category, must be excluded from any other category.
Two events are mutually exclusive if they cannot occur at the same time.
An example is tossing a coin once, which can result in either heads or tails, but not both.
EXAMPLE: eye color.

28. 1-13
Ordinal level
Ordinal level: involves data that may be arranged in some order, but differences between data values cannot be determined or are meaningless.
EXAMPLE: During a taste test of 4 colas,
cola C was ranked number 1,
cola B was ranked number 2,
cola A was ranked number 3,
cola D was ranked number 4.
Rankings (1st, 2nd, 3rd, etc)
Grades (A, B, C, D. F)
Evaluations Hi, Medium, Low

29. 1-13
Interval level
Interval data have meaningful intervals between measurements, but there is no true starting point (zero).
Variables or measurements where the difference between values is measured by a fixed scale.

30. Interval level For example,
For example,
When we measure temperature (in Fahrenheit), the distance from is same as distance from The interval between values is interpretable. Because of this, it makes sense to compute an average of an interval variable, where it doesn't make sense to do so for ordinal scales. But note that in interval measurement ratios don't make any sense - 80 degrees is not twice as hot as 40 degrees
However 0 degrees (in both scales) cold as it may be does not represent the total absence of temperature

32. 1-13
Ratio level
Ratio level: the interval level with an inherent zero starting point. Differences and ratios are meaningful for this level of measurement.
EXAMPLES: money, heights of students.
A measurement such as 0 feet does make sense, as it represents no length.
Furthermore 2 feet is twice as long as 1 foot. So ratios can be formed between the data.

33. www.hndit.com Level of data Nominal Data may only be classified
Classification of
students by
district
Ordinal
Data are ranked
Your rank for
this course
module
Interval
Meaningful
difference
between values
Temperature
Ratio
0 point &
ratio
Number of
study hours

34.
Picture of sampling breakdown
SAMPLING BREAKDOWN

35. SAMPLING…….
STUDY POPULATION
SAMPLE
TARGET POPULATION

36. Types of Samples Probability (Random) Samples Simple random sample
Types of Samples
Probability (Random) Samples
Simple random sample
Systematic random sample
Stratified random sample
Cluster sample
Two general approaches to sampling are used in social science research. With probability sampling, all elements (e.g., persons, households) in the population have some opportunity of being included in the sample, and the mathematical probability that any one of them will be selected can be calculated. With nonprobability sampling, in contrast, population elements are selected on the basis of their availability (e.g., because they volunteered) or because of the researcher's personal judgment that they are representative. The consequence is that an unknown portion of the population is excluded (e.g., those who did not volunteer). One of the most common types of nonprobability sample is called a convenience sample – not because such samples are necessarily easy to recruit, but because the researcher uses whatever individuals are available rather than selecting from the entire population.
Because some members of the population have no chance of being sampled, the extent to which a convenience sample – regardless of its size – actually represents the entire population cannot be known

37. Basic Methods of Sampling
Random Sampling
Selected by using chance or random numbers
Each individual subject (human or otherwise) has an equal chance of being selected
Examples:
Drawing names from a hat
Random Numbers

38. Random Sampling The “pick a name out of the hat” technique
The “pick a name out of the hat” technique
Random number table
Random number generator
A simple random sample of n pieces of data from a population of data is collected in such a manner such that every sample has an equal chance of being selected and included in the sample. The data is chosen randomly using either a random number table or a piece of software called a random number generator.
Hawkes and Marsh (2004)
Sarah DiCalogero - Statistical Sampling

39. Simple Random Sample
Every subset of a specified size n from the population has an equal chance of being selected
Math
Alliance
Project

40. Simple random sampling

41.
Systematic Sampling
This is a form of random sampling, involving a system. Every nth item is selected throughout the list.
Not fully random and therefore there is a possibility of bias.

42. Basic Methods of Sampling
Systematic Sampling
Select a random starting point and then select every kth subject in the population
Simple to use so it is used often

43. Systematic Sampling www.hndit.com All data is sequentially numbered
Every nth piece of data is chosen
A simple random sample of n pieces of data from a population of data is collected in such a manner such that every sample has an equal chance of being selected and included in the sample. The data is chosen randomly using either a random number table or a piece of software called a random number generator.
Hawkes and Marsh (2004)
Sarah DiCalogero - Statistical Sampling

44. Systematic Sample www.hndit.com
Every kth member ( for example: every 10th person) is selected from a list of all population members.
Math
Alliance
Project

45. Stratified random sample
Stratified random sample
In this method all the people or items in the sampling frame are divided into ‘categories’ which are mutually exclusive. Within each level a simple random sample is selected.
Within the categories the samples are random.
But the categories are not clear.

46. Basic Methods of Sampling
Stratified Sampling
Divide the population into at least two different groups with common characteristic(s), then draw SOME subjects from each group (group is called strata or stratum)
Results in a more representative sample

47. Stratified Random Sample
Stratified Random Sample
The population is divided into two or more groups called strata, according to some criterion, such as geographic location, grade level, age, or income, and subsamples are randomly selected from each strata.
Math
Alliance
Project

48. Stratified Sampling Data is divided into subgroups (strata)
Stratified Sampling
Data is divided into subgroups (strata)
Strata are based specific characteristic
Age
Education level
Etc.
Use random sampling within each strata
All of the data is divided into distinct subgroups or strata, based on a specific characteristic such as age, income, education level, and so on. All members of a stratum share the specific characteristics. Random samples are drawn from each stratum.
Hawkes and Marsh (2004)
Sarah DiCalogero - Statistical Sampling

49.
Cluster sampling
Clusters are formed by breaking down the area to be surveyed into smaller areas a number of which are selected by random methods for survey. Within the selected clusters are chosen by random methods for the survey.

50. Basic Methods of Sampling
Basic Methods of Sampling
Cluster Sampling
Divide the population into groups (called clusters), randomly select some of the groups, and then collect data from ALL members of the selected groups
Used extensively by government and private research organizations
Examples:
Exit Polls

51. Cluster Sampling www.hndit.com Data is divided into clusters
Usually geographic
Random sampling used to choose clusters
All data used from selected clusters
The entire population of data s divided into pre-existing segments called clusters. Most often these clusters are geographic. Then clusters are randomly selected, and every member of each selected cluster is included in the sample. NOTE: Used often by government agencies and private research organizations.
Hawkes and Marsh (2004)
Sarah DiCalogero - Statistical Sampling

52. Cluster sampling www.hndit.com Section 1 Section 2 Section 3 Section 5

53. Sampling Relationships
Random Sampling
Cluster Sampling
Stratified Sampling
It is important to note that when using either cluster or stratified sampling you actually have a multi-step sampling process. You need to first divide the data into clusters or strata (depending on the method chosen) and then use random sampling. For cluster sampling you use random sampling to determine the clusters and in stratified sampling you use random sampling to determine the data chosen within each strata.
Sarah DiCalogero - Statistical Sampling

54. Cluster Sample www.hndit.com
The population is divided into subgroups (clusters) like families. A simple random sample is taken of the subgroups and then all members of the cluster selected are surveyed.
Math
Alliance
Project

55. Example 1: Sampling Methods
In a class of 18 students, 6 are chosen for an assignment
Sampling Type
Example
Random
Pull 6 names out of a hat
Systematic
Selecting every 3rd student
Stratified
Divide the class into 2 equal age groups. Randomly choose 3 from each group
Cluster
Divide the class into 6 groups of 3 students each. Randomly choose 2 groups
Convenience
Take the 6 students closest to the teacher
Now we are going to go through a some examples of how to choose 6 students out of a class of 18 by the 5 different sampling methods. After going through the slides I am going to poll the class to ask what method they would choose and why.
Sarah DiCalogero - Statistical Sampling

56. Example 2: Utilizing Sampling Methods
Determine average student age
Sample of 10 students
Ages of 50 statistics students 18 21 42 32 17 19 22 25 24 23 20 29 36 27
In the second example we are going to determine the average age of 2 classes of statistics students by taking a sample of ten students using 3 of the 5 different sampling methods. We will then compare the average ages calculated using each of the 5 methods. We are also numbering the data – for example data point 1 is 18, data point 2 is 21, data point 3 is 42 and so on.
Sarah DiCalogero - Statistical Sampling

57. Example 2 – Random Sampling
Example 2 – Random Sampling
Data Point Location
Corresponding Data Value 35 25 48 17 37 19 14 47 24 4 32 33 34 23 3 42
Mean
25.1
Random number generator
For the random sampling example we used the random number generator to find 10 data points between 1 and 50. We then used these data “points” to find the corresponding student age.
Sarah DiCalogero - Statistical Sampling

58. Example 2 – Systematic Sampling
Example 2 – Systematic Sampling
Data Point Location
Corresponding Data Value 5 17 10 22 15 18 20 21 25 30 35 40 27 45 23 50
Mean
20.8
Take every
data point
5th
For the systematic sampling example since we needed 10 samples and had 50 data points we took (50/10) or every 5th data point.
Sarah DiCalogero - Statistical Sampling