1 Descriptive statistics: Measures of dispersion Mary Christopoulou Practical Psychology 1 Lecture 3

2 Key terms Standard Deviation= τυπική / “σταθερή” απόκλιση από τον μέσο όρο Variance/ variability = Διακύμανση Range = Εύρος

3 Descriptive statistics: statistical measures that summarize and communicate the basic characteristics of a distribution

4 2 Types of Descriptive Statistics Measures of central tendency: measures that communicate the degree to which scores are centred in a distribution Measures of dispersion: measures that communicate the degree to which scores are spread out in a distribution

5 Measures of central tendency Mean Median Mode Measures of dispersion Range Interquartile range Variance Standard Deviation 2 Types of Descriptive Statistics

6 Measures of dispersion They show: how far from the center the data tend to range/ spread. the extent to which scores in a distribution differ from each other How large are the differences between individual scores? How much variability is there in the data?

7 An example N of chocolate cookies consumed by 10 girls and 10 boys: Girls = 5, 3, 9, 12, 10, 12, 4, 9, 7, 2 (mean = 7.3) Boys = 6, 7, 7, 8, 8, 8, 7, 7, 8, 7 (mean = 7.3) Which of the two means is a more accurate reflection of underlying data? 2 issues: consistency of data: smaller amount of variability indicates a greater consistency of the data (and vice versa) Accuracy/reliability of measure of central tendency

8 Example cntd… Girls = 5, 3, 9, 12, 10, 12, 4, 9, 7, 2 (mean = 7.3) Boys = 6, 7, 7, 8, 8, 8,7, 7, 8, 7 (mean = 7.3) Girls’ mean is based on scores with greater variability, and Boys’ mean is based on scores with smaller variability. Mean for Boys is a more accurate reflection of underlying data, as it is based on a sample that is more consistent (from one score to the next).

9 Range Range is the difference (distance) between the highest and lowest value in the data  Can be calculated for all levels of measurement, apart from the nominal level.

10 RANGE example RANGE = = , 2, 4, 9, 5, 7, 1, 2

11 How to calculate the range 1. Put scores from lowest through highest 2. Find the highest value and the lowest value of the data set 3. Subtract the lowest score from the highest score Example: 22, 25, 30, 42, 88, 102 Range is 102 – 22 = 80

12 Range Used as a very quick (“rough”) method Quite inefficient – only the smallest and largest values are used Excessively vulnerable to outliers (extreme scores)

13 Inter Quartile Range The IQR (Inter Quartile Range) is not affected by extreme scores Divide the data into 4 equal parts (Quartiles) delete the extreme quarters of data and measure range of middle 50%

How to find the IQR: 1. Put scores in order 2. Delete the extreme quartiles IQR= = 3 3 3

15 Variance and Standard Deviation Two measures of variability that tell us how much scores are spread around a mean. Note: a mean of 50 could indicate that most scores are between 48-52, or could indicate that most scores are between 40 and 80!!

16 Variance (or Variability) It is the degree to which scores are spread around the mean It involves the average square deviation of each value from the mean of the values It is the average error between the mean and the observations made.

17 Example Which set of scores are more spread out? Set A: 40, 40, 50, 60, 60  mean=50 Set B: 40, 49, 50, 51, 60  mean=

18 If we subtract each score from its group mean we can see that: 4 of the scores in set A are 10 units away from the mean, whereas only 2 scores in set B are 10 units from the mean. Thus A has the greater variance!!

19 Variance The most efficient of the measures of dispersion Only valid for interval & ratio scales Vulnerable to outliers The larger the value of the Variance, the more each score is “distant” from the mean. The smaller the Variance, the closer each score is, to the mean.

20 The Standard Deviation (SD) Recall that calculation of the Variance involves squaring the deviation scores. This means the Variance value is much larger than the actual deviation of scores from the mean. Therefore, the variance value is not reported often. Instead, report the Standard Deviation.

21 Standard Deviation Is the average amount by which scores in a distribution differ from the mean  Shows the average distance of the data from the mean  It is a measure of how well the mean represents the data.  Is a measure of the degree of dispersion of the data from the mean.

22 Interpreting SD… large SD = scores are far from the mean (so, the mean is not an accurate representation of the data). small SD = scores are closer to the mean (scores are more clustered around the mean). E.g. 3 data sets, each has an average of , 0, 14, 14  SD = , 6, 8, 14  SD = , 6, 8, 8  SD = 1 The 3rd set has a much smaller standard deviation than the other 2 because its values are all close to 7.

23 Interpreting SD using our example… Recall that mean = 5, SD = 2.00 in our example “Boundary at minus one SD” = = 3 “Boundary at plus one SD” = = 7 These boundaries indicate how much scores are spread around the mean. Thus, the majority of scores in our sample are between 3 and 7 (this is true)

24 Important! In scientific reports (including your lab write-ups) it is important to report both a measure of central tendency and a measure of dispersion Mean & SD for normally distributed data Median & IQR for skewed data

25 Reporting Results APA format Q. what is the mean and standard deviation? A. (M = 5.00, SD = 2.00).

26 Small Revision Descriptive Statistics Measures of Central Tendency Measures of Dispersion Range Interquartile Range Variance Standard Deviation