Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Understanding Measures of Central Tendency and Dispersion in Descriptive Statistics, Study notes of Psychology

This document from psy 301 course introduces the concept of descriptive statistics, focusing on measures of central tendency (mode, median, mean) and dispersion (range, variance, standard deviation). Why these measures are important and how they help in summarizing data sets. It also discusses the limitations of each measure and provides examples to illustrate their usage.

Typology: Study notes

Pre 2010

Uploaded on 08/18/2009

koofers-user-xc2
koofers-user-xc2 🇺🇸

10 documents

1 / 15

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
1
Descriptive Statistics
Psy 301
Psy 301
Two most frequently u sed types of descriptive
statistics:
- measures of ce ntral tendency
- measures of va riability
Why do we want to kn ow this about our data?
Descriptive statistics a re used to summarize a set of da ta into just
a few numbers that represent the entire data set.
Descriptive vs. Inferential Statistics
Descriptive Statistics
Describe the data in h and
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 la rger (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
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff

Partial preview of the text

Download Understanding Measures of Central Tendency and Dispersion in Descriptive Statistics and more Study notes Psychology in PDF only on Docsity!

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. 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