Measures of central tendency and dispersion

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# Measures of central tendency and dispersion

## Measures of central tendency and dispersion

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##### Presentation Transcript

1. Measures of central tendency and dispersion

2. Measures of central tendency • Mean • Median • Mode • ie finding a ‘typical’ value from the middle of the data.

3. You need to be able to: • Explain how to calculate the mean, median and mode • State the strengths and weaknesses of mean, median and mode • This could include saying which one you would use for some data e.g. 2, 2, 3, 2, 3, 2, 3, 2, 97 - would you use mean or median here?

5. Measures of Dispersion • Measures of ‘spread’ • This looks at how ‘spread out’ the data are. • Are the scores similar to each other (closely clustered), or quite spread out?

6. Range and standard deviation • The range is the difference between the highest and lowest numbers. What is the range of … • 3, 5, 8, 8, 9, 10, 12, 12, 13, 15 • Mean = 9.5 range = 12 (3 to 15) • 1, 5, 8, 8, 9, 10, 12, 12, 13, 17 • Mean = 9.5 range = 16 (1 to 17) • Example from Cara Flanagan, Research Methods for AQA A Psychology (2005) Nelson Thornes p 15

7. Standard deviation • Standard deviation tells us the average distance of each score from the mean. • 68% of normally distributed data is within 1 sd each side of the mean • 95% within 2 sd • Almost all is within 3 sd

8. Example • Mean IQ = 100, sd = 15 • What is the IQ of 68% of population (ie what is the range of possible IQs)? • Between what IQ scores would 95% of people be?

9. Another example • Sol scores 61% in the test. His mum says that’s rubbish. Sol points out that the mean score in class was 50%, with an sd of 5. Did he do well? • What if the sd was only 2? • What if sd was 15?

10. Advantages and disadvantages I used Cara Flanagan’s (2005) Research Methods for AQA A Psychology Nelson Thornes in preparing these slides.

11. Standard Deviation • Standard deviation (SD) is a statistical measure of the amount the results vary from the mean. • There are 2 formulas that can be used to work out the standard deviation: • Formula 1 • Formula 2: S= √∑d² n Formula 1 is used to calculate the SD where the whole population has been used. S= √∑d² n-1 Formula 2 is used to calculate the SD where part the population has been used. This is the formula used most often.

12. Standard Deviation • Example - Test scores from 10 students: • 85, 86, 94, 95, 96, 107, 108, 108, 109, 112 • Calculate the mean of the data • Mean = 100 • Square the differences between each score and mean: Test score d d2 85 -15 225 (85-100 = -15 -152 = 225) 86 -14 196 94 -6 36 ... .. .. 109 9 81 112 12 144 • Sum of d2 = 900 • N = number of scores = 10, so n-1 = 9 • Substitute into the formula and we get 10, because 900 divided by 9 is 100 and the square root of 100 is 10

13. Standard Deviation • Example 2: Calculate the standard deviation of the following scores: 12, 10, 8, 4, 18, 8 • Mean = 10 • Subtract each score from the mean to give deviation of score from mean • (-2, 0, 2, 6, -8, 2) • Square each deviation • (4, 0, 4, 36, 64, 4) • Add the squares of the deviation together = 112 • Count the number of scores = 6, and then subtract 1 • Divide the sum of the squares of the deviation by the number of scores minus 1 = 112/5 = 22.4 (called the variance) • Square root the variance = standard deviation = 4.73

14. Remember… If we were just to look at the mean score without any further information, this data would be misleading. THEREFORE, YOU NEVER REPORT THE MEAN WITHOUT THE STANDARD DEVIATION. Large standard deviations suggest that there is a large variance in the scores, and possibly 1 or 2 scores are distorting the mean Small standard deviations suggest that there are no large variations in the scores distorting the mean.

15. Task • Scores showing the number of seconds taken by participants to complete a task without alcohol (condition 1) and with alcohol (condition 2) Condition (1) Condition (2) 3 3 4 6 4 6 5 9 5 9 5 9 6 11 6 13 7 15 • Calculate the: • Mean • Mode • Median • Standard dev • For both conditions

16. Task Answers • Calculate the mode, median, mean and standard deviation for each condition Mode Median Mean Std Dev • Condition 1 5 5 5 1.22 secs • Condition 2 9 9 9 3.71 secs • So what do the measures of central tendency and dispersion tell us?

17. Task Interpretation • The means tell us… • Participants who had consumed alcohol (condition 2) took nearly twice as long to complete the task as the participants who had not consumed alcohol (condition 1) • The standard deviations tell us… • Spread of scores in condition 2 was much greater than the spread in condition 1. • Scores in condition 1 are clustered around the mean, whilst those in condition 2 are more widely spread out

18. Task Conclusions • What do the findings indicate generally? • Alcohol increases the average time taken to perform a task • It exaggerates differences in task performance • slows some people down a lot and others hardly at all

19. Task • Complete hand outs to ensure that you are confident in working out the measures of central tendency and the standard deviation