Please enable JavaScript.
Coggle requires JavaScript to display documents.
Chapter 4 - Variability - Coggle Diagram
Chapter 4 - Variability
4-1 Introduction to Variability
Goal is to measure the amount of variability for a set of scores (or distribution)
Variability
provides a quantitative measure of the differences between scores in a distribution
describes the degree to which scores are spread out or clustered together
can be viewed as measuring predictability, consistency, and even diversity
2 purposes of a good measure of variability:
Variability describes the distribution scores - tells whether the scores are clustered close together or spread out over a large distance.
Variability measures how well an individual score (or group of scores) represents the entire distribution).
usually defined in terms of distance
The Range
the distance covered by the scores in a distribution (from smallest to largest)
alternative definition used when scores are measurements of a continuous variable
when scores are all whole numbers based on a discrete variable, the range can be obtained by:
seldom used in formal descriptions of variability because of its failure to reveal the typical distance among common scores
The Interquartile Range
range of scores that make up the middle 50% of the distribution
based on
quartiles
- a type of percentile rank; one-fourth of a distribution
1st quartile = Q1 - score that has a percentile rank of 25% (25% of scores fall below it)
2nd quartile = Q2 - percentile rank of 50% - the median
3rd quartile = Q3 - score with a rank of 75% (75% of scores fall below it)
4th quartile = Q4 - highest score in the distribution, rank is 100%
IQR (interquartile range)
= distance between th 25th and 75th percentile (Q1 and Q3)
typically used when measuring central tendency with median because both are related to percentiles. IQR can also be used when there are undetermined values and open-ended distributions. It can be used for data measured with an ordinal scale.
4-2 Defining Variance and Standard Deviation
Standard Deviation
most commonly used and most important descriptive measure of variability
uses the mean of the distribution as a reference point
measures variability by considering the distance between each score and the mean
provides a measure of the standard/average distance from the mean
describes whether the scores are clustered closely around the mean or are widely scattered
Deviation
- distance from the mean
Deviation score - difference between a score and the mean
Steps to finding SD
Determine the deviation - *note: there are 2 parts to a deviation score - the sign (+ or -), which tells the direction from mean (above or below) and the number
Calculate mean of deviation scores - add deviation scores, then divide by N - *sum of deviations is always zero
Square each deviation score. Using these squared values, compute the
variance
Variance
= the mean of the squared deviations. It is the average squared distance from the mean.
Take the square root of the variance to obtain the
standard deviation
Standard deviation - the square root of the variance and provides a measure of the standard, or average distance from he mean.
4-3 Measuring Variance and Standard Deviation for a Population
Sum of Squared Deviations
- sum of the squared deviation scores
Formulas to compute SS:
Definitional formula
:
use when have a small group of scores and mean is whole number
Computational formula
:
performs calculations with the scores (not the deviations); therefore, minimizes the complications of decimals and fractions
Final Formulas and Notation
Population Variance - σ2; equals the mean squared distance from the mean.
Population Standard Deviation
- σ; equals the square root of the population variance
4-4 Measuring Variance and Standard Deviation for a Sample
Problem with Sample Variability:
bias
in the direction of underestimating the population value
Formulas for Sample Variance (s2) and SD (s)
Definitional formula:
Computational Formula:
sample variance = estimated population variance
sample standard deviation = estimated population standard deviation
Sample Variability and Degrees of Freedom
degrees of freedom (df)
: determine the number of scores in the sample that are independent and free to vary df= n-1
To calculate sample variance:
Formula for sample standard deviation:
4-5 Sample Variance as an Unbiased Statistic
Biased Statistic
average value of the statistic either underestimates or overestimates the corresponding population parameter
Unbiased statistic
average value of the statistic is equal to the population paramete
4-6 More about Variance and Standard Deviation
In frequency distribution graphs
: identify the position of the mean by drawing a vertical line and labeling it with µ or M; SD is represented by a horizontal line or an arrow drawn from the mean outward for a distance equal to the DS and labeled with σ or s.
Transformations of Scale
sets of scores can be transformed by adding a consonant to each score or by multiplying each score by a constant value
Adding a constant to each score does not change the standard deviation
multiplying each score by a constant causes the standard deviation to be multiplied by the same constant
Variability plays an important role in the inferential process
in the context of inferential statistics, the variance is often classified as
error variance
- term used to indicate that the sample variance represents unexplained and uncontrolled differences between scores
as error variance increases, it becomes more difficult to see any systematic differences or patterns that might exist
variance makes it difficult to get a clear signal from the data
roughly 70% of the scores in a distribution are located within a distance of 1 SD from the mean
almost all the scores (roughly 95%0 are within 2 SD of the mean
Mean and SD are not simply abstract concepts or mathematical equations; they are concrete and meaningful values
