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

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Practical Psychology 1 Lecture 3Descriptive statistics:Measures of dispersionMary Christopoulou

Key termsStandard Deviation= /

Variance/ variability =

Range =

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

2 Types of Descriptive StatisticsMeasures of central tendency:measures that communicate the degree towhich scores are centred in a distribution

Measures of dispersion:measures that communicate the degree towhich scores are spread out in adistribution

2 Types of Descriptive StatisticsMeasures ofcentral tendencyMeanMedianMode

Measures ofdispersionRangeInterquartile rangeVarianceStandard Deviation

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

An exampleN 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

Example cntdGirls = 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).

RangeRange 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.

RANGE exampleRANGE = 9 - 1 = 8123452793, 2, 4, 9, 5, 7, 1, 2

How to calculate the rangePut scores from lowest through highestFind the highest value and the lowest value of the data set Subtract the lowest score from the highest score

Example: 22, 25, 30, 42, 88, 102Range is 102 22 = 80

RangeUsed as a very quick (rough) methodQuite inefficient only the smallest and largest values are used Excessively vulnerable to outliers (extreme scores)

Inter Quartile RangeThe 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%

28241527How to find the IQR:1. Put scores in order 2. Delete the extreme quartiles12245728IQR= 5 - 2 = 333

Variance and Standard DeviationTwo 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!!48 - - - 5240 - - - - - - - - - - - - - - - - - - - - - - - - - - - - 80

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

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

4050604060

40 50 60 49 51

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!!

VarianceThe most efficient of the measures of dispersionOnly valid for interval & ratio scalesVulnerable 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.

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.

Standard DeviationIs the average amount by which scores in a distribution differ from the mean Shows the average distance of the data from the meanIt 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.

Interpreting SDlarge 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 7.0, 0, 14, 14 SD = 70, 6, 8, 14 SD = 56, 6, 8, 8 SD = 1

The 3rd set has a much smaller standard deviation thanthe other 2 because its values are all close to 7.

Interpreting SD using our exampleRecall that mean = 5, SD = 2.00 in our exampleBoundary at minus one SD = 5 - 2 = 3Boundary at plus one SD = 5 + 2 = 7These 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)

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 dataMedian & IQR for skewed data

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

Small RevisionDescriptive StatisticsMeasures of Central TendencyMeasures of DispersionRangeInterquartile RangeVarianceStandard Deviation

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