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Descriptive Statistics
Psy 301 Two most frequently used types of descriptive statistics:
- measures of central tendency
- measures of variability Why do we want to know this about our data? Descriptive statistics are used to summarize a set of data into just a few numbers that represent the entire data set.
Descriptive vs. Inferential Statistics
- Descriptive Statistics
- Describe the data in hand
- Test scores in this class
- Memory spans obtained from 30 subjects
- Time to graduation of WSU students in the freshman class of 2000
- Inferential Statistics
- Infer the nature of a larger (typically infinite) set of data that we don!t have - Scores of all students I might have for 301 in the next 10 years - Memory spans for normal adults - Time to graduation for college students
Example: You open a men!s shoe store Since someone told us that the average shoe size for a man is 9 , we borrow money and buy 1000 pairs of size 9 shoes. What happens? You go broke! Psy 301 Example: You reopen a men!s shoe store You try again, but this time you borrow money and buy 1100 pairs of shoes, 100 of each of size 4 -
What happens? You still go broke! Psy 301 Example: You reopen a men!s shoe store After a few weeks of being open, here is what your stock of shoes looks like. 4 5 6 7 8 9 10 11 12 13 14 You did not take into account that there are not equal numbers of feet of each size.
Mode: the most frequent score
Measures of central tendency Ex: 5 9 7 4 6 8 2 4 1 3 5 1 4 6 9 8 7 5 2 4 1 ordered: 1 1 1 2 2 3 4 4 4 4 5 5 5 6 6 7 7 8 8 9 9 3 2 1 4 3 2 2 2 2 Mode
Psy 301
Mode
Measures of central tendency
- there is no mode when all the scores are different (or there is the same number of many scores)
- sometimes there is more than one mode Psy 301 Fig 4. 3 (Heiman, p. 71 ) Unimodal distribution
Figure 4. 4 (Heiman, p. 71 ) Bimodal distribution Psy 301
Mode
Measures of central tendency
- there is no mode when all the scores are different (or there is the same number of many scores)
- sometimes there is more than one mode Limitation
- does not take into account other scores so comments about distribution may be misleading Psy 301
Median: the middle score (of an ordered set)
Measures of central tendency Ex (odd # of scores): 3 8 11 11 12 13 24 35 46 Median
Mean: the arithmetic average of scores
Measures of central tendency Ex: 3 8 11 11 12 13 24 35 46 48
Mean: X = (!X)
N
X = ( 3 + 8 + 11 + 11 + 12 + 13 + 24 + 35 + 46 + 48 )
X = ( 211 ) = 21. 1
- 1 Mean Psy 301
Mean
Measures of central tendency
- deviation of any score = X-X (keep the sign +/-)
- the sum of deviation scores for any distribution will be zero ( 0 ) Psy 301 Table 4. 1
Mean
Measures of central tendency
- deviation of any score = X-X (keep the sign +/-)
- the sum of deviation scores for any distribution will be zero ( 0 ) Limitation
- susceptible to outliers (really different scores)
Mean: X = (!X)
N
sample
Mean: μ
population Psy 301 What measure of central tendency should you use? 1 ) What type of measurement scale did you use? If Nominal - then you must use the mode If Ordinal - then you can use either the mode or the median If Interval or Ratio - you can use any but the mean is the most typical 2 ) Look at your data. What does the distribution look like? Psy 301 Normal Distribution Low Mean, Mode Median High Scores Frequency
Along with a measure of central tendency , we also need to know the spread of the scores. That is we need a measure of dispersion or variability. Psy 301 Measures of dispersion (variability) A single number that is used to represent how the scores of a data set are spread out. Spread is important because alone, the mean (or median or mode) does not give someone a sufficient idea of the way the data look. Psy 301
Range: the range of scores covered
Measures of dispersion (variability) Range = (highest score - lowest score) Range = 9 - 1 = 8 Ex: 5 9 7 4 6 8 2 4 1 3 5 1 4 6 9 8 7 5 2 4 1
Range
Measures of dispersion (variability) Limitation
- sensitive to outliers
- does not take into account the magnitude of scores between highest & lowest Psy 301
Variance: average squared deviation
from the mean
Measures of dispersion (variability) Why is it squared? Remember that the sum of the deviations of scores for any distribution is always zero ( 0 ). Squaring the deviations makes all the numbers positive. Variance = !
( X " X )^2
N Psy 301
Variance
Measures of dispersion (variability) Definitional formula: ! SX^2 = ( X^ "^ X^ ) #^2 N Computational formula: ! SX^2 = X^2 " ( (^) # X )^2 N
N
Computing variance & standard deviation using the computational formula ! 0 0 30 120 6 36 5 25 4 16 4 16 3 9 3 9 2 4 2 4 1 1 X X^2 !^ (!X)^2 = 30^2 = 900 SX =^ X^ (^2) " ( # X )^2
N
! N SX^2 = X^2 " (^ X ) #^2
_N
N_ N = 10 ! SX^2 = 120 " (^90010) 10 =^ 120 " 90 10 =^ 30 10 =^3 ! SX = 120 " (^90010) 10 =^ 120 " 90 10 =^ 30 10 =^3 =^ 1. Psy 301
Population vs Sample
Population: the infinitely large group of all possible scores that could be obtained if the behavior of every individual of interest in a particular situation could be measured (Heiman, 2003 ). Sample: a relatively small subset of a population, intended to represent the population; a subset of the complete group of scores found in any particular situation (Heiman, 2003 ). Measures of dispersion (variability) Psy 301 Measures of dispersion: population
Population Variance
! " (^) X^2 = $( X^ #^ μ)^2 N
Population Standard Deviation
! " (^) X = $( X # μ)^2 N
Sample estimates of population parameters All of the previous formulas (except population ones) have been for samples where we are interested in just describing that particular sample. Measures of dispersion (variability) BUT, usually we measure the behaviors of a sample because we want to get an idea of how the larger population would behave. Thus we use the sample statistics to estimate the population parameters. Psy 301 Sample estimates of population statistics Measures of dispersion (variability) When we use the sample statistics to estimate the population statistics, we change the formulas a little. Variance: ! sX^2 = X^2 " (^ X ) #^2 N
N " 1 Standard Deviation: ! sX = X^2 " ( # X )^2 N
N " 1 lower case Psy 301 Sample estimates of population statistics Measures of dispersion (variability) N- 1 is used because samples will tend to underestimate the real population variance. When N- 1 is used, these are called unbiased estimators. Variance: ! sX^2 = X^2 " (^ X ) #^2 N
N " 1 Standard Deviation: ! sX = X^2 " (^ X ) #^2 N
N " 1
