Variability stats ch 4

More about Variance and Standard Deviation

Unbiased: estimate of a population parameter Average value of statistic is equal to parameter Average value sues all possible samples of a particular size n. Biased: estimate of a population parameter Systematically overestimates or underestimates the population parameter

Formulas for Population Variance and Standard Deviation

Variance= Sum of Squared Deviations/Number of Scores Deviation score= X-u Sum of Squared Deviation= (E)(X-u)2 Variance = (E)(X-u)2/N SS(sum of squares) is the sum of the squared deviations of scores from the mean; SS=(E)(X-u)2 Two equations for computing SS *Sum of Squares, the number of the variance. Deviation is hidden but, actually means (Deviation scores squared) u=mean

Variance and standard Deviation for a Sample -example

We know that the sample variance and standard deviation can underestimate the population variance and SD if we use n. To correct this problem, we used n-1 instead of n and made the value of the variance and SD a little bit bigger. It does not mean, however, every statistics using n-1 is exactly the same with parameters. Rather, the average of all sample variances can produce an accurate estimate of population variance. This accurate estimate of parameter is thought to be unbiased, while the sample statistics using n, for example, is thought to be biased (because they can underestimate the parameters). Using n-1 is just an effort to get unbiased statistics as close to parameters as possible. *when you're asked in a question that states this group is a sample, you need to consider n-1 formula, continuous variable

Standard Variance Example

X X-u (X-u)2 3 0 0 3 0 0 5 2 4 1 -2 4 4 1 4 3 0 0 2 -1 1 3 0 0 N=8 u= (E)X/N = 24/8 =3 Variance= (E)(X-u)2/N = 10/8= 1.25 Standard Deviation =square root of variance square root 1.25 = 1.118

Variance and standard Deviation for a Sample

*To make a sample variance and SD as unbiased estimates for those of the corresponding population.. SS is computed as before.. Formula for Variance has n-1 rather than n in the denominator Notation uses s instead of (o) variance of sample= s2 = SS/n-1 -unbiased standard deviation of sample= s= square root of SS/n-1

Range example

62,85,97,76,68,83 highest score:97 lowest score:62 Range = 97-62=35 *The range as a measure of variability of a distribution is sensitive to extreme values in the distribution. So, if there is any extreme value (outlier) in a distribution, the range would not be a good index of variability of the dataset.

Learning Check 3 True or False

A biased statistic has been influenced by researcher error? TRUE On average, an unbiased sample statistic has the same value as the population parameter? TRUE

Variability

A quantitative measure of the differences between scores Describes the degree to which the scores are spread out or clustered together.

Learning check 2

A sample of four scores has SS=24. What is the variance? Four scores is n, so 4-1= Sample variance SS/df-->24/4 = SS/n-1 so, 24/3 = 8 The variance is 8.

True or False 2

A sample systematically has less variability than a population TRUE The standard deviation is the distance from the Mean to the farthest point of the distribution curve? FALSE (most popular measure of variability is SD)

Purposes of Measure of Variability

Describe the distribution Measure how ell an individual score represents the distribution.

Two Formula for SS (1) the two formulas are equivalent) Definitional Formula

Find each deviation score (X-u) Square each deviation score (X-u)2 Sum up the squared deviations SS=(E)(X-u)2 if the mean (u) is a decimal then we consider Computation Formula

Calculation of Variance

Find the deviation (distance from the mean) for each score, add the deviations and compute the avg. (dead end the value is always 0---> Square each deviation--->Find the avg. of the squared deviations(called "variance')--.Take the square root of the variance--->The standard deviation or standard distance from the mean

Variance and Inferential Statistics

Goal of inferential statistics is to detect meaningful and insignificant patterns in research results. Variability in the data influences how easy it is to see patterns High variability obscures patterns that would be visible in low variability samples Variability is sometimes called Error Variance

Standard Deviation and Variance for a Sample

Goal of inferential statistics: Draw general conclusions about population by estimating parameters using sample statistics Based on limited information from a sample. Samples differ from a population: Samples have "less variability" (sampling error) Computing the Variance and SD for a sample in the same way as for a population would give a biased estimate for the population values *less variability-Systematic difference of sample variability from that of a corresponding population. Sample variability is always smaller than pop. variability-ITS BIASED

The Interquartile Range (IQR)

Is the distance from the 75th percentile (Q3; the 3rd quartile) to the 25th percentile (Q1; 1st quartile). Therefore, this range covers 50% of scores in a distribution. Ex. 1,2,2,|3,3,3,|3,4,4,|5,6,7 n=12 Q3=4.5 Q1=2.5 Interquartile range = Q3-Q1= 4.5-2.5=2

Standard Deviation and Variance for a Population

Most common and most important measure of variability A measure of the standard, or average, distance from the mean. Describes whether the scores are clustered closely around the mean or are widely scattered. Calculation differs for population and samples.

Population Variance: Formula and Notation

Notation: Lowercase Greek letter sigma is used to denote the standard deviation of population (o) Because the standard deviation is the square root of the variance, we write the variance of a population as (o)2 Formula: Variance = (E)(X-u)2/N =(o)2 Standard Deviation= square root of (E)(X-u)2/N = (o)

Degrees of Freedom

Population Variance: Mean is known-Deviations are computed from a known mean. Sample variance as estimate of population-Population mean is unknown, Using sample mean restricts variability. Degrees of Freedom: Number of scores in sample that are independent and free to vary. Degrees of freedom (df) = n-1

Three Measures of Variability

Range, Standard Deviation and Variance

Two Formula for SS (2) the two formulas are equivalent) Computational Formula

Square each square and sum the squared scores Find the sum of scores, square it, divide by N Subtract the second part from the first SS= (E)X2 - (EX)2/N Example: 1,2,3,4 EX=10 EX2= 1+4+9+16=30 SS= 30 -(10)2/4 =30-25 =5

Developing the Standard Deviation

Step One: Determine the Deviation, Deviation is the distance from the mean. Deviation score = X - (u) (meaning the magnitude of it) Step Two: Calculate mean of Deviations Deviations sum to 0 because M is balance point of distribution The Mean Deviation will always equal o; another method must be found (E)(X-u)2 (getting rid of the negative sign) (E)(X-M)/N

Developing the Standard Deviation (2)

Step Three: Get rid of + and - in Deviations Square each deviation score Compute the Mean Squared Deviation, known as Variance **Population variance equals the mean squared deviation. Variance is the average squared distance from the mean. Variability is now measured in squared units Step Four:Goal: to compute a measure of standard distance of the scores from the mean Variance measures the average squared distance from the mean;not quite on goal. Correct for having squared all the differences by taking the square root of the variance. Standard Deviation = square root of Variance

Learning Check

Sum of squared deviation scores (SS sum of squares) Standard distance a score from the mean SD Average deviation of a score from the mean zero 0 Average squared distance of a score from the mean - Varaince

True or False

The computational & definitional formulas for SS sometimes give different results? FALSE NEVER If all the scores in a dataset are the same, the SD is equal to 1.00 FALSE (should be 0 because they are the same.

Variability definition

The degree to which how scores in a distribution are spread or grouped together. usually variability is defined in terms of distance, and there are multiple numerical measures showing variability of a distribution.

The Range

The distance covered by the scores in a distribution (from smallest value to highest value) For continuous data, real limits are used range = Xmax. - Xmin. Based on two scores, not all the data Considered a crude unreliable measure of variability.