Spss introductory session data entry and descriptive stats
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Transcript of Spss introductory session data entry and descriptive stats
SPSS & Quantitative Data AnalysisKulbir Singh Birak
SPSS is a computer program for analysing quantitative data.
This can range from basic descriptive statistics such as the mean, mode, median and range to powerful tests of significance (So whether we accept or reject a hypothesis).
What the data looks like, and what that means if anything.
What is SPSS?
You can access SPSS on the vast majority of PC’s at UCS, in these labs, the Waterfront PC’s and the library PC’s
Additionally, if you wish you can borrow a copy of SPSS to install on your own home PC or laptop. There are 16 copies in the library you just need to borrow the disc and input the license code that comes with it (license’s do come to an end and when they do you can just come and borrow a new version of SPSS or attain a new license code)
Over night loan only or you can bring your laptops in and do there and then
Windows version only, no Apple version
SPSS Access
Overview
Why do numbers matter in research design?
Numbers allow you to do two basic things:- Count how often
“something” happens
- Count how big an issue “something” is
Overview
Once you can count the extent (how often) and nature (for quantitative research a numerical descriptor of an attribute) you can already do some pretty important things. You can answer questions such as:
How common is an issue? For instance, are black
children over-represented in care? Are black adults over-represented in psychiatric hospital?
How serious is a particular issue? Or how is it distributed within a sample? For instance, how serious
are the concerns about children in families allocated a social worker?
OverviewOnce you can count stuff you can start to answer other important and interesting questions, for instance:
Students may often come to you with various questions about SPSS and difficulties that they are having
If you are lucky enough to catch them early on a lot of unnecessary frustration and stress about analysing data can be avoided.
The most important thing a student can do before they even consider methodology, methods or analysis is to have a clear research question/aim and hypotheses in place that conceptualise and operationalise the variables they wish to study.
SPSS and Quantitative Data
Some Basic Definitions
A variable is the “thing” that you’re interested in studying e.g. depression, gender differences, social deprivation,
specific crime rates, levels of emotionality (how emotional someone
is) or different types of food!
• Things like depression, gender differences, social deprivation, specific crime rates, levels of emotionality and food type, etc. are called “variables” because they vary.● Some people are more depressed than others● Some people are men, and others are women● Some Social policies may be more successful than others● We may see different crimes committed in different
contexts, areas ● Some people are less emotional than others● Food types can range from pizza to hamburgers to filet
mignon, or might be Thai, Ethiopian, Polish or American cuisine, etc., etc.
TO “CONCEPTUALISE” A VARIABLE MEANS TO MAKE CLEAR WHAT YOU MEAN BY THE VARIABLE….
• For example, for the variable “food type,” you need to be clear about whether you mean
• (1) vegetarian or meat, OR• (2) breakfast, lunch or dinner foods, OR• (3) Ethiopian, Thai or American foods, OR • (4) something else!
TO “OPERATIONALISE” A VARIABLE IS TO DECIDE HOW YOU WILL MEASURE IT
• For example, if the variable you’re interested in is depression:● Will you ask people to rate themselves, and if
so, on what sort of a scale?● Alternatively, will you measure depression by
facial expression? By some behaviour that you observe? In some other way?
TO “OPERATIONALISE” A VARIABLE IS TO DECIDE HOW YOU WILL MEASURE IT
• If the variable you’re studying is intelligence & you don’t think Exam scores are a good measure of intelligence, what measure WILL you use?
• Asking these sorts of questions is completing the process of “operationalising” your variables.
• Conceptualisation & Operationalisation are necessary for a Quantitative approach
Exploratory Descriptive Causal/RelationshipExploratory research is undertaken when few or no previous studies exist. The aim is to look for patterns, hypotheses or ideas that can be tested and will form the basis for further research.
Typical research techniques would include case studies, observation and reviews of previous related studies and data.
Data from exploratory studies tends to be qualitative.
Expands on the Exploratory
Descriptive research can be used to identify and classify the elements or characteristics of the subject, e.g. number of days youth offenders remained out of trouble.
Quantitative techniques are most often used to collect, analyse and summarise data.
Causal and Relationship research focuses on being able to predict/hypothesise cause and effect between observed behaviours, or relationships between aspects of behaviour/society/crime rates.
The idea is that Causal and Relationship research is moving a step beyond descriptive research and the quantitative data collected can be used and analysed in a manner that allows the researcher to infer a significant effect/difference or relationship
TYPES OF QUANTITATIVE RESEARCH
Aims and Objectives
• The Quantitative approach sets out at the start of a study with a research question and a hypothesis/prediction
• Hypotheses are formal statements of predictions derived from evidence from earlier research and/or theory.
• The null hypothesis (H0) is a statement of ‘no difference/effect/change’ between the variables
• The experimental hypothesis (H1) is a statement of difference/relationships between variables
QUANTITATIVE DESIGNS AND HYPOTHESES
• Experimental Hypothesis: Students who study for tests in study groups will score significantly better on their exams than students who did not study in study groups
• Null Hypothesis: There will be no significant difference in exam results between students who do and do not study in study groups
EXAMPLE OF HYPOTHESIS
This clarity in the question and hypothesis can make life markedly easier for yourselves and the student in the long run.
However, I appreciate that this is not always the easiest/or will not be the case for you more often than not.
So what I will be covering with you today is a brief introduction to the SPSS interface and as to how we would go about doing the initial basics of data entry and beginning to explore descriptive data.
If we can I’ll also take you through examples of some basic significance testing (otherwise I’ll put up so available)
A light but important session. Going over the basics of how to input data, label your
variables so it is clear and how to create codebooks. It’s all about building up your confidence with the
interface, and developing good practise. It’s about doing the basics so as to avoid confusion
later on, e.g. inputting the data correctly for different types of analysis.
Data Entry and Descriptives
Hopefully should be familiar with the idea of descriptive data.
As the name suggests they are what we use to describe the data we have.
There’s no point in knowing that the IQ scores between two groups are significantly different if we don’t have a way of describing the scores, and the difference.
Measures of central tendency: Mean, mode, median etc.
Measures of dispersion: Standard deviation etc.
Descriptive Stats
Levels of Measurement
In 1946 Stevens proposed a theory of scales of measurement. Nominal data (lowest level of
measurement) Ordinal data (unable to differentiate
points on scale) Interval data (points on scale equal
distance apart) Ratio data (equal distance between
points on scale)
Nominal
Provides the least exact information Participants are placed in categories
Data that is categorical e.g. gender, colours, shoe type, play behaviour
Variable must fit into one category Measure of frequency
Numbers may be used but only as category labels
Central tendency is described using the mode Data is represented using a frequency table
or bar chart
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
Simplest true scale, orders measurements along a continuum Represent rank position in a group e.g. 1st, 2nd, 3rd …10th No information on difference between positions
Central tendency is described in terms of the median
Dispersion can be measured using the range or inter-quartile range (middle 50% of the distribution)
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!
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 (e.g. income), possible to calculate ratios so also has the interval property (e.g. someone earning £20,000 earns twice as much as someone who earns £10,000) (Fielding & Gilbert 2000:15)
Difference between interval and ratio usually not important for statistical analysis (Fielding & Gilbert 2000:15)
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. 120 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 (Papadopoulos, 2001).
If you are still a little worried about your understanding of Quantitative Data please see the Key Information Handout in the Folder. By David Bowers (Learning Development) A reasonable summary of information about
quantitative data. Data types, appropriate measures of central
tendency etc.
Key Information Handout
Everything we do today is about good practice.
Following the steps today, and developing correct inputting skills, will save you lots of problems and heartache later.
SPSS is fussy when it comes to the way data is entered.
Importance of Good Practice
As SPSS is a Quantitative Data analysis software you often have to reduce information down to a numerical state
A Codebook allows you to keep a record of these reductions and decisions
A record of your own. Separate from SPSS. Electronic or on paper. A list of variables, full names, and how you
have coded data.
Codebook
The codes you give data to allow SPSS to analyse it.
You can’t enter text so some variables need to be converted.
E.g. Gender: Female may become 1, Male may become
2.
Relationship Status: Single may become 1, Married 2, Divorced 3, Widowed 4…
Coding
SPSS is fussy when it comes to the names you give variables.
Can’t give them a full description in the main view.
So you can give detailed labels in the special variable view.
Along with a codebook it helps keep the information clear.
Labelling
Available on email that was circulated to you all
File: Data Entry Exercise 1 - Optimism Data We’ll be creating a codebook, setting up
SPSS according to the codebook, and then entering the data.
1st Exercise
Good habits
Create a new Folder on your Desktop
Right-click on Desktop> New > Folder > “SPSS”
New Data Folder
Start>All Programs>IBM SPSS Statistics 19.
Depending on version may have a slightly different name.
GIVE IT TIME SPSS IS RENOWNED FOR TAKING AN AGE TO OPEN UP – CLICKING AGAIN ONLY SLOWS IT DOWN MORE AS IT’LL THEN TRY TO OPEN ANOTHER SPSS WINDOW
Open SPSS
Open SPSS
Optimism Scale data from 4 participants
Firstly, we are going to prepare a codebook
Coding Data
Optimism Hand-out
Rules for naming of variables Variable names:
must be unique (i.e. each variable in a data set must have a different name);
must begin with a letter (not a number); cannot include full stops, spaces or other
characters (!, ? * "); cannot include words used as commands by
SPSS (all, ne, eq, to, le, lt, by, or, gt, and, not, ge, with)
Coding Data
Optimism scale items op1 to 4 Enter number circled 1 (strongly disagree)
to 5 (strongly agree)
Coding Data
Now we have a codebook to keep things clear we can set up SPSS so it is ready for the data.
SPSS has 3 views: Data, Variable and Output.
By switching to Variable we can define the variables we need.
Creating a data file and inputting data
Defining Variables
Variable View
Naming Variables
Decimals
Labels
Values
Enter the relevant value and label as per your codebook, then click add. When all have been entered, click OK
Define the meaning of the values used in the codebook (Gender) and click add for each.
Values
Values
When entering likert data always use the limits of the scale (1-5) even if you know that participants may not have entered some responses. You also need to decide whether you are going o just enter the range or every labeled point.
Values
Data comes in different types. Categorical (Nominal in SPSS) Ordinal Scale/Interval (Scale in SPSS) Different types/measures suit different tests, different measures of central tendency, different forms of visualisation. Makes knowing what type of data you have KEY for successful data analysis.
Measures
Measures
Scale refers to interval/ratio level of measurement - There is some debate about data type in relation to likert data … for our purposes, leave this as Scale
Nominal refers to catergorical
Measures
Now you have the variables set up ready for the data you can start to enter the actual data
Go to the Data View
Inputting Data According to the Codebook
Inputting Data According to the Codebook
Inputting Data According to the Codebook
Saving the File
Saving the File
You’ve saved the data so now it is ‘safe’ You can have a play around with it and try
a few different things. Delete a case Insert a case between existing cases Delete a variable Insert a variable between existing variables Try during the workshop/at home so you
get more confident with SPSS.
Playing around with the data
Available on LearnUCS.
Different experimental designs require a different style of inputting.
The structure you use will be different between Repeated (Within-Group) and Independent (Between-Group) experimental designs.
Use the wrong structure and the analysis will fall down. It will be meaningless at best.
2nd Exercise: Inputting Repeated and Independent
Measures
So, to recap Repeated Measures. The same participants
experience all treatments/are in all the groups/conditions.
If you wanted to investigate the effect of music on taking an IQ test participants would experience the no music condition, and the music condition.
Hopefully with some counterbalancing.
Repeated Measures
Repeated Measures
Repeated Measures
Again to recap.
Participants are split. One group will experience one treatment/be in one group/condition.
Another group will experience the other.
Each condition will have a unique, non-shared, set of participants.
Independent
Independent
Independent
Independent
Independent
Independent
Independent
A quick trick to show you. Good for those who aren’t fond of a
screen full of numbers. If you have coded your variables
correctly there is a button you can press that will make the numbers in your data view appear as the names coded.
For example the 1’s and 2’s for gender could appear as Male and Female.
Labelling Trick
Data Entry Exercise 1 – Optimism Data Input Data Entry Exercise 2 – Repeated and Independent Extra Data Entry Exercises
Exercise 3 – Giving electric shocks Exercise 4 – Shooting people
We’ve gone through 1 and 2 here. Try them on your own. 3 and 4 for extra practice. Make sure you are comfortable with data input, coding and labelling.
Exercises
The theory and step-by-step guide will be covered in the slides following immediately below. If you complete the first exercise move onto exercise 2.
Descriptive Exercise 1: survey.savThe data is from a survey of staff about stress and emotions.Generate the frequencies for 1) marital status and 2) level of education
Descriptive Exercise 2: staffsurvey.savThe data is from a staff survey with likert scales for agreement and importance of factors.Generate appropriate descriptive statistics to answer the following questions:
(a) What percentage of the staff in this organisation are permanent employees? (Use the variable employstatus.)(b) What is the average length of service for staff in the organisation? (Use the variable service.)(c) What percentage of respondents would recommend the organisation to others as a good place to work? (Use the variable recommend.)
Lab Exercises
The theory and step-by-step guide will be covered in the slides following immediately below. If you complete the first exercise move onto exercise 2.
Descriptive Exercise 1: survey.savThe data is from a survey of staff about stress and emotions.Generate the frequencies for 1) marital status and 2) level of education
Descriptive Exercise 2: staffsurvey.savThe data is from a staff survey with likert scales for agreement and importance of factors.Generate appropriate descriptive statistics to answer the following questions:
(a) What percentage of the staff in this organisation are permanent employees? (Use the variable employstatus.)(b) What is the average length of service for staff in the organisation? (Use the variable service.)(c) What percentage of respondents would recommend the organisation to others as a good place to work? (Use the variable recommend.)
Lab Exercises
When you are trying to find your descriptive stats you need to make sure you use the right ones.
Certain types of data/measure, suit certain types of measures of central tendency and dispersion.
Use the wrong ones and your description of the results will be confusing, wrong and won’t match your inferential statistics.
Types of Variables & Descriptives
Also known as Nominal variables in SPSS. Data that has been classified and categorised. So gender, a participant will belong to a
particular category of gender. Marital Status. Anything that you can create a discrete
classification of. You can even take a scale variable like age, and force it into categories (18 and under, 18 – 25, 25 – 35 etc.).
Categorical Variables
Measure of Central tendency to use for Categorical data is the mode.
Frequency of occurrence or amount. So using gender as an example you would
use the mode. 2 of the sample might be male, and 8 female. Mode = Female. 20% male, 80% female
Categorical
In SPSS you should use the Frequency option when you want the descriptive stats for a categorical variable.
Go to Descriptive Exercise 1 on LearnUCS.
Categorical and Frequency
Save survey.sav to your SPSS folder on the Desktop from LearnUCS
Have a look at survey.sav questionnaire from LearnUCS
Open survey.sav dataset
Descriptive Exercise 1 - Survey
Survey Questionnaire
Frequencies
Frequencies
Frequencies
Frequency Output
Frequency Output
This is where graphs and the results from tests (descriptive and inferential) will appear.
Also notes about when you have saved and opened files too.
If you want to keep what is in the output you must save it specifically.
Saving the data/variable will not save what is in the output, and vice versa.
Output pages
Aside from Categorical measures we also have Ordinal Scale/Interval (sometimes know as ratio too) These are also generally known as continuous
variables. Usually the mean or median are the measures
of central tendency used, and the standard deviation, or error, the measure of dispersion.
Other measures
Ranked or ordered data. Sometimes Likert scales.
Has some similarity to categorical data (You might consider grade brackets to be categories; A, B, C, D, etc).
But importantly they are ranked, so there is meaning to the position. A is better than B, B better than C and so on.
The median is used here. Central point with an equal amount above/below.
Ordinal
The median is used here. Central point with an equal amount above/below So if you had a collection of grades… 20 people had an A 10 had a B 10 had a C 10 had a D Then B would be the median grade, as 20 people
had higher, and 20 people had lower.
Ordinal
Imagine we wished to find the median for the highest educational level attained by a population
In descriptive exercise 1 (survey) we would click on ‘Analyze’
Using Explore to See the Median
Using Explore to See the Median
Select ‘Descriptive Statistics’ and then ‘Explore’ from the Drop-down menus
Using Explore to See the Median
1. When the below box opens move ‘highest educ completed’ from the left pane to the ‘Dependent List’ section
2. Click on ‘Statistics’ and choose ‘Outliers’ and ‘Continue’ 3. Click on ‘Plots’ and choose ‘Histograms’ and ‘Normality Plots with tests’ and ‘Continue’
4. Click on ‘OK’
Using Explore to See the Median
The resulting ‘Output’ in the Output window will show you a number of descriptive stats.We can see the median is 4 for the ‘highest educ completed’ which means ‘some additional training’ is the median for the highest education completed for 439 participants who took part in the survey.
Interval – a scale with artificial limits, no true zero, and usually some form of cap.
Intervals are of equal size. IQ scores for example. Ratio – has a true zero, constant intervals and
potentially little or no cap. So timing scores on a task for example. SPSS doesn’t really differentiate between the two. Basically if it is a form of score it is likely to be scale.
Scale
The mean is the normal measure of central tendency, and the measure of dispersion the standard deviation.
So 5 people take a maths test. They score 10, 20, 18, 12 and 5. The average would be 13 (total/number of
cases)
Scale
In SPSS we just need the descriptive option, rather than the frequency option.
So for example if we wished to find the mean and standard deviation for ‘age’, ‘total optimism’, ‘total mastery’, ‘total perceived stress’ and ‘total perceived control of internal states’ (PCOISS), for participants who answered the survey we are using for exercise 1.
Scale Descriptives
Descriptives
Descriptives
Descriptives
Descriptives Output
Sometimes information will be left out of a questionnaire, or the value lost, but you will still need to conduct an analysis.
What happens if someone doesn’t fill in the age box on a questionnaire?
Rather than get rid of all their data you can use the ‘Exclude cases pairwise’ option.
It excludes the case (person) only if they are missing the data required for the specific analysis. They will still be included in any of the analyses for which they have the necessary information.
Missing Data
Exclude cases listwise A more extreme option. If the participant is missing any data then
this option should remove them entirely from the analysis.
A matter of judgement as to which to use.
Missing Data
Descriptive Exercise 1 – Survey Descriptive Exercise 2 – Staff Survey
Exercises
Adapted from Green, J. & D’Oliveira, M. (1999). Learning to use statistical tests in psychology. Buckingham, UK: Open University Press.
Differences ?
Categorical & FrequencyData? Relationships ?
How many Independent variables?
START
Within orBetween
participants in each condition?
Two or more
Parametric: Unrelated t-test
Non-param:Mann Whitney
Between
How many experimental conditions?
One
Factorial Within Subjects (Repeated Measures) ANOVA
Within
Factorial Mixed Design (Split-Plot) ANOVA
Both True
Between
Factorial Between Groups ANOVA
3 or more
Within orBetween
participants in each condition?
Two
Within orBetween
participants in each condition?
Parametric: Non-param:Oneway FriedmanWithin Ss or(Repeated Page’s Lmeasures) Trend TestANOVA
Within Between
Parametric: Non-param:Oneway Kruskal-Between Wallis orGroup JonckheereANOVA Trend Test
Parametric: Non-Param: Related Wilcoxont-test
Within
Parametric: Non-param:Pearson's r Spearman's r
Flowchart for choosing basic statistics
Summarising Univariate Data?
Descriptive statistics(mean, standard deviation,variance, etc)
1 or 2 sample Chi-square
Within
McNemar
Between
Coolican, H. (2014). Research Methods and Statistics in Psychology (6th ed.). Hove, UK: Psychology Press. A good introduction to the quantitative statistics incorporated in
the social sciences. A comprehensive coverage of the statistics covered in research methods at this level in a clear and comprehensive format.
Pallant, J. (2013). SPSS: Survival Manual (5th ed.). Maidenhead, UK: Open University Press A textbook that is of help with the statistical programme SPSS
whatever your level, as it takes you through the analysis in a step-by-step clear and concise manner that allows you to learn while you put into practice.
Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.). London, UK: Sage An easy to engage with text that covers research methods and
statistics in a fashion that makes it easy to read and follow.
Recommended Reading
You can use the below link to access the UCS library page that has some useful videos showing how to use SPSS http://libguides.ucs.ac.uk/c.php?g=264784&p=1954991
There is also a course that you can do (set up by Jen Versey our Psychology technician and David Mullett from the library support team) https://www.coursesites.com/webapps/Bb-sites-course-creatio
n-BBLEARN/courseHomepage.htmlx?course_id=_383196_1
There is always the IBM SPSS guide that you can access through the help option in SPSS as a starting point.
Web Resources
Descriptive Statistics
Descriptive statistics – are statistics that describe data. They essentially summarise the data.
They can be either numerical or graphic Numerical statistics come in 2 forms
Measurement of central tendency Measurement of dispersion
Measure of Central Tendency
Three measures of central tendency/ score, which we use is dependent on our level of measurement. They are;
Mean Arithmetic average/mean. Sum of all scores divided by
the number of scores Median
The score that falls in the exact centre of the distribution (middlemost score)
Mode The most common/frequently occurring score
‘the mean’ Formula for the mean is_ Σxx = N_x = the meanΣ = the sum ofx = the scoresN = the number of scores in set Advantages
Powerful statistic used in estimating population parameters for significant differences and correlations. Most sensitive, and works at an interval level.
Disadvantages Can be overly sensitive causing it to easily distort due to outlier values
‘the median’
The measure of central tendency for ordinal data Shorthand may be Guildford’s (1956) Mdn It is the central value of a set A formula used to find the median is N + 1k = 2 For odd number data sets this will reveal the central number For even number data sets this will reveal the two points of data that
the median falls between When you have a number of values the same in the data set you can
use the same method although it is not strictly correct. However, luckily for us as social scientists there are statistical packages that will take care of this for us
‘the mode’
The measure of central tendency for nominal scale data. We are unable to calculate mean and median with this type of data, but we can see what occurred most often/highest frequency
There can be two modes, which we call bi-modal Advantages
Most typical, unaffected by extremes, can be more informative than mean with discrete scales
Disadvantages Does not account for differences between values, can’t be used in
estimates of population parameters, not all that useful for small sets of data, for bi-modal two modal values reported, difficult to estimate accurately when data grouped into class intervals
Measures of Spread/Dispersion
High Variability
Low Variability
‘the range’
Report of the top/highest value and the bottom/lowest value
To calculate what the range is (the difference between) you subtract the lower value from the higher value and add 1
Advantage Includes extremes, easy to calculate
Disadvantages Can be distorted by extremes, can be unrepresentative of the
distribution. Doesn’t tell us whether values close to spaced out from mean
‘the interquartile and semi-interquartile range’
The interquartile range allows us a better insight into how values fall in relation to the central tendency
Instead of the full range, the interquartile range represents the distance between the central 50%, removing the bottom and top 25%. The values are known as the 1st and 3rd quartiles or the 25th and 75th percentiles
Interquartile range
Q1 M Q33 3 4 5 6 8 10 13 14 16 19 The interquartile range is: Q3 – Q1
Semi-interquartile is half of that: Q3 – Q1
2 Advantages
Representative of central group of values, useful for ordinal data
Disadvantages No account of extremes, inaccurate where there are large
class intervals
Standard deviation and variance
These estimate from a sample how the values of a population are distributed
Standard deviation provides us with an average score telling us how different the scores are from the mean
Formula for standard deviation (std, SD, stdev)
)(1
2
nXx
s 1
n
s2d Or