Understanding variability, variance and standard deviation

Measuring variability

The variability of a distribution refers to the extent to which scores are spread or clustered. Variability provides a quantitative value to the extent of difference between scores. A large value refers to high variability. The aim of measuring variability is twofold:

  1. Describing the distance than can be expected between scores;

  2. Measuring the representativeness of a scores for the whole distribution.

The range of a measurement is the distance between the highest and lowest score. The lowest score should be subtracted from the highest score. However, the range can provide a wrong image when there are extreme values present. Thus, the disadvantage of the range is that it does not account for all values, but only for the extreme values.

Variance and standard deviation

The standard deviation (SD) is the most frequently used and most important measure for spread. This measurement uses the mean of the distribution as comparison point. Moreover, the standard deviation uses the distance between individual scores and the mean of the data set. By using the standard deviation, you can check whether individual scores in general are far away or close to the mean. The standard deviation can be best understood by means of four steps:

  1. First, the deviation of each individual score to the mean has to be calculated. The deviance is the difference between each individual score and the mean of the variable. The formula is:

\[deviation\: score = x - µ\]

  • x: individual score of x
  • μ: mean of the variable
  1. In the next step, calculate the mean of the deviation scores. This can be obtained by adding all deviations scores and dividing the sum by the number of deviation scores (N). The deviation scores are combined always zero. Before computing the mean, each deviation score should be placed between brackets and squared.

\[mean\:of\:the\:deviation\:scores = \frac{\sum{(x-\mu)}}{N}\]

  • x: individual score of x
  • μ: mean of the variable
  • N: number of deviation scores
  1. Next, the mean of the squared sum can be computed. This is called the variance. The formula of the variance is:

\[σ^2= \frac{\sum {(x-μ)^{2}}}{N}\]

  • σ2: squared sum or variance
  • x: individual score of x
  • μ: mean of the variable
  • N: number of deviation scores
  1. Finally, draw the square root of the variance. The result is the standard deviation. The final formula for the standard deviation is thus:

\[σ= \sqrt {\frac{\sum {(x-μ)^{2}}}{N}}\]

  • σ: standard deviation
  • x: individual score of x
  • μ: mean of the variable
  • N: number of deviation scores

Often, the variance is a large and unclear number, because it comprises a squared number. It is therefore useful and easier to understand to compute and present the standard deviation.

In a sample with n scores, the first n-1 scores can vary, but the last score is definite. The sample consists of n-1 degrees of freedom (in short: df).

Systematic variance and error variance

The total variance can be subdivided into 1) systematic variance and 2) error variance.

  • Systematic variance refers to that part of the total variance that can predictably be related to the variables that the researcher examines.

  • Error variance emerges when the behavior of participants is influenced by variables that the researcher does not examine (did not include in his or her study) or by means of measurement error (errors made during the measurement). For example, if someone scores high on aggression, this may also be explained by his or her bad mood instead of the temperature. This form of variance can not be predicted in the study. The more error variance is present in a data set, the harder it is to determine if the manipulated variables (independent variables) actually are related to the behavior one wants to examine (the dependent variable). Therefore, researchers try to minimize the error variance in their study.

Statistics: Magazines for encountering Statistics

Statistics: Magazines for encountering Statistics

Startmagazine: Introduction to Statistics
Stats for students: Simple steps for passing your statistics courses

Stats for students: Simple steps for passing your statistics courses

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How to triumph over the theory of statistics (without understanding everything)?

Stats of students

  • The first years that you follow statistics, it is often a case of taking knowledge for granted and simply trying to pass the courses. Don't worry if you don't understand everything right away: in later years it will fall into place, and you will see the importance of the theory you had to know before.
  • The book you need to study may be difficult to understand at first. Be patient: later in your studies, the effort you put in now will pay off.
  • Be a Gestalt Scientist! In other words, recognize that the whole of statistics is greater than the sum of its parts. It is very easy to get hung up on nit-picking details and fail to see the forest because of the trees
  • Tip: Precise use of language is important in research. Try to reproduce the theory verbatim (i.e. learn by heart) where possible. With that, you don't have to understand it yet, you show that you've been working on it, you can't go wrong by using the wrong word and you practice for later reporting of research.
  • Tip: Keep study material, handouts, sheets, and other publications from your teacher for future reference.

How to score points with formulas of statistics (without learning them all)?

  • The direct relationship between data and results consists of mathematical formulas. These follow their own logic, are written in their own language, and can therefore be complex to comprehend.
  • If you don't understand the math behind statistics, you don't understand statistics. This does not have to be a problem, because statistics is an applied science from which you can also get excellent results without understanding. None of your teachers will understand all the statistical formulas.
  • Please note: you will probably have to know and understand a number of formulas, so that you can demonstrate that you know the principle of how statistics work. Which formulas you need to know differs from subject to subject and lecturer to lecturer, but in general these are relatively simple formulas that occur frequently, and your lecturer will likely tell you (often several times) that you should know this formula.
  • Tip: if you want to recognize statistical symbols, you can use: Recognizing commonly used statistical symbols
  • Tip: have fun with LaTeX! LaTeX code gives us a simple way to write out mathematical formulas and make them look professional. Play with LaTeX. With that, you can include used formulas in your own papers and you learn to understand how a formula is built up – which greatly benefits your understanding and remembering that formula. See also (in Dutch): How to create formulas like a pro on JoHo WorldSupporter?
  • Tip: Are you interested in a career in sciences or programming? Then take your formulas seriously and go through them again after your course.

How to practice your statistics (with minimal effort)?

How to select your data?

  • Your teacher will regularly use a dataset for lessons during the first years of your studying. It is instructive (and can be a lot of fun) to set up your own research for once with real data that is also used by other researchers.
  • Tip: scientific articles often indicate which datasets have been used for the research. There is a good chance that those datasets are valid. Sometimes there are also studies that determine which datasets are more valid for the topic you want to study than others. Make use of datasets other researchers point out.
  • Tip: Do you want an interesting research result? You can use the same method and question, but use an alternative dataset, and/or alternative variables, and/or alternative location, and/or alternative time span. This allows you to validate or falsify the results of earlier research.
  • Tip: for datasets you can look at Discovering datasets for statistical research

How to operationalize clearly and smartly?

  • For the operationalization, it is usually sufficient to indicate the following three things:
    • What is the concept you want to study?
    • Which variable does that concept represent?
    • Which indicators do you select for those variables?
  • It is smart to argue that a variable is valid, or why you choose that indicator.
  • For example, if you want to know whether someone is currently a father or mother (concept), you can search the variables for how many children the respondent has (variable) and then select on the indicators greater than 0, or is not 0 (indicators). Where possible, use the terms 'concept', 'variable', 'indicator' and 'valid' in your communication. For example, as follows: “The variable [variable name] is a valid measure of the concept [concept name] (if applicable: source). The value [description of the value] is an indicator of [what you want to measure].” (ie.: The variable "Number of children" is a valid measure of the concept of parenthood. A value greater than 0 is an indicator of whether someone is currently a father or mother.)

How to run analyses and draw your conclusions?

  • The choice of your analyses depends, among other things, on what your research goal is, which methods are often used in the existing literature, and practical issues and limitations.
  • The more you learn, the more independently you can choose research methods that suit your research goal. In the beginning, follow the lecturer – at the end of your studies you will have a toolbox with which you can vary in your research yourself.
  • Try to link up as much as possible with research methods that are used in the existing literature, because otherwise you could be comparing apples with oranges. Deviating can sometimes lead to interesting results, but discuss this with your teacher first.
  • For as long as you need, keep a step-by-step plan at hand on how you can best run your analysis and achieve results. For every analysis you run, there is a step-by-step explanation of how to perform it; if you do not find it in your study literature, it can often be found quickly on the internet.
  • Tip: Practice a lot with statistics, so that you can show results quickly. You cannot learn statistics by just reading about it.
  • Tip: The measurement level of the variables you use (ratio, interval, ordinal, nominal) largely determines the research method you can use. Show your audience that you recognize this.
  • Tip: conclusions from statistical analyses will never be certain, but at the most likely. There is usually a standard formulation for each research method with which you can express the conclusions from that analysis and at the same time indicate that it is not certain. Use that standard wording when communicating about results from your analysis.
  • Tip: see explanation for various analyses: Introduction to statistics
Statistics: Magazines for understanding statistics

Statistics: Magazines for understanding statistics

Startmagazine: Introduction to Statistics
Understanding data: distributions, connections and gatherings
Understanding reliability and validity
Statistics Magazine: Understanding statistical samples
Understanding distributions in statistics
Understanding variability, variance and standard deviation
Understanding inferential statistics
Understanding type-I and type-II errors
Understanding effect size, proportion of explained variance and power of tests to your significant results
Statistiek: samenvattingen en studiehulp - Thema
Statistics: Magazines for applying statistics

Statistics: Magazines for applying statistics

Applying z-tests and t-tests
Applying correlation, regression and linear regression
Applying spearman's correlation - Theme
Applying multiple regression
Statistiek: samenvattingen en studiehulp - Thema

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