# Sampling and Basic Descriptive Statistics

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Sampling and Basic Descriptive Statistics
Sampling and Basic Descriptive Statistics. Basic concepts and Techniques. Lecture 6 Leah Wild

Overview Sampling In Quantitative Research
Basic Descriptive Statistics And Graphical Representation Of Data Quantification, Variables And Levels Of Measurement

Sampling In Quantitative Research
Total Population Representative sample Probability Sampling Non-Probability Sampling Sample Size

Total Population The total collection of units, elements or individuals that you want to analyse. These can be countries, lab-rats, light bulbs, university students, banks, residents of a particular area, regional health authorities etc. The population for a study of infant health might be all children born in the U.K. in the 1980's.

Sample A sample is a group of units selected from a larger group (the population). By studying the sample it is hoped to draw valid conclusions about the larger group. Using example for study of infant health the sample might be all babies born on 7th May in any of the years. samples selected because the population is too large to study in its entirety. Important that the researcher carefully and completely defines the population, including a description of the members to be included

Representative sample
A sample whose characteristics correspond to, or reflect, those of the original population or reference population To ensure representativeness, the sample may be either completely random or stratified depending upon the conceptualized population and the sampling objective (i.e., upon the decision to be made). A thorny issue in the social sciences- is it possible to achieve?

Probability Sampling A probability provides a quantitative description of the likely occurrence of a particular event. A probability sampling method is any method of sampling that uses some form of random selection. In order to have a random selection method, you must set up some process or procedure that assures that the different units in your population have equal probabilities of being chosen (Clark 2002: 37).

Most Common Types of Probability Sampling
Simple Random Sampling Stratified Random Sampling Systematic Random Sampling Cluster Or Multistage Sampling

Simple Random Sampling
where we select a group of subjects (a sample) for study from a larger group (a population). Each individual is chosen randomly and each member of the population has an equal chance of being included in the sample. Every possible sample of a given size has the same chance of selection; that is, each member of the population is equally likely to be chosen at any stage in the sampling process. (Easton & Mc Coll 2004). A lottery draw is a good example of simple random sampling. A sample of 6 numbers is randomly generated from a population of 45, with each number having an equal chance of being selected.

Stratified Random Sampling
Often factors which divide up the population into sub-populations (groups / strata) measurement of interest may vary among the different sub-populations. This has to be accounted for when we select a sample from the population to ensure our sample is representative of the population. This is achieved by stratified sampling. A stratified sample is obtained by taking samples from each stratum or sub-group of a population. Suppose a farmer wishes to work out the average milk yield of each cow type in his herd which consists of Ayrshire, Friesian, Galloway and Jersey cows. He could divide up his herd into the four sub-groups and take samples from these (Easton and Mc Coll 2004).

Systematic Random Sampling
Systematic sampling, sometimes called interval sampling, means that there is a gap, or interval, between each selection. Often used in industry, where an item is selected for testing from a production line (say, every fifteen minutes) to ensure that machines and equipment are working to specification. Alternatively, the manufacturer might decide to select every 20th item on a production line to test for defects and quality. This technique requires the first item to be selected at random as a starting point for testing and, thereafter, every 20th item is chosen. used when questioning people in surveys eg market researcher selecting every 10th person who enters a particular store, after selecting a person at random as a starting point; interviewing occupants of every 5th house in a street, after selecting a house at random as a starting point. If researcher wants to select a fixed size sample. In this case, it is first necessary to know the whole population size from which the sample is being selected. The appropriate sampling interval, I, is then calculated by dividing population size, N, by required sample size, n, as follows: If a systematic sample of 500 students were to be carried out in a university with an enrolled population of 10,000, the sampling interval would be: I = N/n = 10,000/500 =20

Cluster Or Multistage Sampling
Cluster sampling is a sampling technique where the entire population is divided into groups, or clusters, and a random sample of these clusters are selected. All observations in the selected clusters are included in the sample. every element should have a specified (equal) chance of being selected into the final sample. typically used when the researcher cannot get a complete list of the members of a population they wish to study but can get a complete list of groups or 'clusters' of the population Cheap, easy economical method of data collection.

Non-Probability Sampling
Main Types Convenience/ opportunity/accidental sampling. Purposive/ judgemental sampling Quota sampling Snowball sampling

Convenience/ opportunity/accidental sampling.
volunteer samples Sometimes access through contacts or gatekeepers ‘easy to reach’ population.

Purposive/ judgemental sampling
Involves selecting a group of people because they have particular traits that the researcher wants to study e.g. consumers of a particular product or service in some types of market research My own questionnaire research on ‘New-Age’ Travellers.

Quota sampling widely used in opinion polls and market research.
Interviewers given a quota of subjects of specified type to attempt to recruit. eg. an interviewer might be told to go out and select 20 male smokers and 20 female smokers so that they could interview them about their health and smoking behaviours .

Snowball sampling Involves two main steps.
Identify a few key individuals Ask these individuals to volunteer to distribute the questionnaire to people who know and fit the traits of the desired sample (e.g. my research on Travellers)

Sample Size In general, the larger the sample size (selected with the use of probability techniques) the better. The more heterogeneous a population is on a variety of characteristics (e.g. race, age, sexual orientation, religion) then a larger sample is needed to reflect that diversity. (Papadopoulos 2003) Response rates vary on the type of surveys (e.g. mail surveys, telephone surveys). Response rates under 60 or 70 per cent may compromise the integrity of the random sample. (ibid)

Basic Descriptive Statistics And Graphical Representation Of Data
Can be divided into two types Descriptive. Inferential Some authors suggest a third type ‘Associative’ (Downey 1975)

Descriptive Statistics
Statistics which describe attributes of a sample or population. includes measures of central tendency statistics (e.g., mean, median, mode), frequencies, percentages. minimum, maximum, and range for a data set, variance etc. organise and summarise a set of data

Inferential Statistics
Used to make inferences or judgments about a larger population based on the data collected from a small sample drawn from the population. Eg Exit polling used during US elections to determine how the population of voters voted A key component of inferential statistics is the calculation of statistical significance of a research finding. used to determine whether changes in a dependent variable are caused by an independent variable (Clark 2004) (HOMEWORK- WHAT ARE SOME OF THE PROBLEMS ASSOCIATED WITH THESE KIND OF STATISTICS?

Quantification, Variables And Levels Of Measurement
Rowntree (2000) distinguishes between category variables and quantity variables. Category variables can be nominal or ordinal. Quantity variables can be discrete or continuous.

Examples: Nominal Data
Type of Bicycle Mountain bike, road bike, chopper, folding,BMX. Ethnicity White British, Afro-Caribbean, Asian, Chinese, other, etc. (note problems with these categories). Smoking status smoker, non-smoker

Ordinal Data A type of categorical data in which order is important.
Class of degree-1st class, 2:1, 2:2, 3rd class, fail Degree of illness- none, mild, moderate, acute, chronic. Opinion of students about stats classes- Very unhappy, unhappy, neutral, happy, ecstatic!

Discrete Data Continuous Data
Only certain values are possible (there are gaps between the possible values). Implies counting. Continuous Data Theoretically, with a fine enough measuring device. Implies counting.

Relationships between Variables. (Source. Rowntree 2000: 33)
Quantity Category Continuous (measuring) Discrete (counting) Ordinal Nominal Ordered categories Ranks.

Quantification, Variables And Levels Of Measurement
Fielding and Gilbert (2000: 15) distinguish between four levels of measurement. Nominal Ordinal. Interval Ratio.

Interval and ratio variables
According to Fielding & Gilbert (2000) these are often used interchangeably, and incorrectly by social scientists.(pg15) Interval, ordered categories, no inherent concept of zero (Clark 2004), we can calculate meaningful distance between categories, few real examples of interval variables in social sciences. (Fielding & Gilbert 2000:15) Ratio. A meaningful zero amount (eg income), possible to calculate ratios so also has the interval property (eg someone earning £20,000 earns twice as much as someone who earns £10,000).(ibid) Difference between interval and ratio usually not important for statistical analysis (ibid).

Interval variables- Examples
Fahrenheit temperature scale- Zero is arbitrary- 40 Degrees is not twice as hot as 20 degrees. IQ tests. No such thing as Zero IQ IQ not twice as intelligent as 60. Question- Can we assume that attitudinal data represents real, quantifiable measured categories? (ie. That ‘very happy’ is twice as happy as plain ‘happy’ or that ‘Very unhappy’ means no happiness at all). Statisticians not in agreement on this.

Ratio variables-Examples
Can be discrete or continuous data. The distance between any two adjacent units of measurement (intervals) is the same and there is a meaningful zero point (Papadopoulos 2001) Income- someone earning £20,000 earns twice as much as someone who earns £10,000. Height Unemployment rate- measured as the number of jobseekers as a percentage of the labour force (ibid).

IMPORTANT! SEE ‘TYPES OF DATA REVISION SLIDES’ ON MY WEBSITE FOR EXTRA INFORMATION ON TYPES OF DATA

Frequencies and Distributions
Frequency-A frequency is the number of times a value is observed in a distribution or the number of times a particular event occurs. Distribution-When the observed values are arranged in order they are called a rank order distribution or an array. Distributions demonstrate how the frequencies of observations are distributed across a range of values.

Example Look at the distribution below:
This distribution shows the recorded ages of patients receiving treatment for heart disease in the Stroud district. There are 50 observed values. We can easily see how often each value occurs. What is the frequency of the following values, 79; 81; 94? What is the range of this distribution?(r = h – l ). What is the mode? What is the median? From this distribution we can also tell that most of the values tend to cluster around the middle of the range. 62 64 65 66 68 70 71 72 73 74 75 76 77 78 79 80 81 82 83 85 86 87 88 89 90 92 94 96

Two elements to a distribution
Scale with a number of values -(Usually arrange the scores from the highest to lowest). Corresponding observations- Tally up the scores, convert them into frequencies.

Types of Distribution Frequency distribution Class Intervals
Relative (Proportional or percentage distributions) Cumulative distributions.

Frequency Distributions
Shows number of cases having each of the attributes of a particular variable. Divided into two types Ungrouped distribution-scores not collapsed into categories, each score represented as a separate values Grouped distribution. Scores collapsed into categories so that several scores are presented together as a group. Groups usually referred to as a class interval.

Relative (proportional or percentage) distributions
The proportion of cases in the whole distribution observed at each score or value.

Cumulative distribution.
The number of cases up to and including the scale value. Can appear in grouped or ungrouped format. Cumulative relative distribution for any particular value is the the total up to, and including, that value