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Descriptive Statistics 1. Mean, median and mode.

Extracts from this document...

Introduction

Descriptive Statistics 1. Mean, median and mode.

Introduction

At this stage I want to emphasize the practical relevance of averaging, using the discussion of the mean to illustrate the use of mathematical notation (that was introduced last week), and warn you about some of the possible pitfalls of relying on the average without looking at the pattern of values from which it was calculated. First, however, I will attempt to answer the good question, “why bother learning statistical notation”, that seems to crop up every year.

Why learn mathematical notation?

PSY107 aims to provide not just a recipe book for doing statistics problems in isolation, but aims to leave you with some skills that have general relevance. These include computer literacy, the ability to explore data, critical thinking, and a degree of independence in tackling statistical issues in Psychology and elsewhere. The routine tools that I and many of my colleagues use to do statistics are not algebra and equations, but (often computerized) graphing and data analysis methods. I think that many simple statistical concepts can be communicated using graphs and plain English. Why, if many psychologists do not spend their time writing μ and σ is it necessary to get to grips with the basics of statistical notation?

This question has several answers. First, mathematical language is logical, rigorous, and compact. Mathematical notation can also be used to represent the precise relationship between different statistical concepts.

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Middle

Definition of the mean

The arithmetic mean is the sum of the values divided by the number of values. This is shown below using mathematical notation. It is best to get to grips with the symbols while the statistical concepts they represent (e.g. mean average) are simple and familiar.

Sample mean vs. population mean

μ, the population mean, is the mean derived from the entire population under study. Population is a word with a somewhat elastic meaning, but generally it is up to you, the experimenter, to define your population. It might be all the people in the UK, all the people who shop at KwikSave, or all the lecturers in the Newcastle University Psychology Department. With large populations, it is often impractical to find μ.

image16.png

image02.png, the sample mean, is calculated from a representative sample of the population. This is usually done by selecting individuals from the population at random to avoid sampling bias. You get sampling bias when all the members of the population under study do not stand an equal chance of being measured.

image17.png

If you wanted to estimate the mean height of people in the UK, it would be stupid to do all your measuring in primary schools. This is an extreme example, but more realistically, suppose you wanted to get a representative 1000 people to complete a questionnaire on social attitudes. If you did the survey by telephone, your sample would be biased towards telephone owners. If you called between 9 and 5, your sample would be biased towards people without day jobs.

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Conclusion

The Median.

The median is the value that divides the distribution of values exactly in half. To find the median, sort or rank the values and find the middle value (if there are is an odd number of values) or else the mean of the central two values (if there is an even number of values). It is possible to estimate the median from histograms. Use the information on number of scores to estimate the position of the middle score.

Fig. 7

image11.png

Estimate the position of the “middle person” on the income axis using the information on the frequency axis. Here there are 18 lecturers, so the median income is at the estimated position between 9 and 10. This is simpler to estimate from a cumulative frequency polygon.

The median average can be more representative than the mean in skewed distributions (e.g. annual income, or National Lottery winnings). Remember to look at the data when you calculate the median average

Fig. 8

image12.png

The Mode

The mode is the score or category that has the greatest frequency. The modal average can be used with nominal data. As with all other averages, look at the data when you calculate the mode.

Fig. 9

image13.png

Is a three dimensional representation sensible?

Mike Cox

4.

Version 1

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