Stats for Behavioral Sciences: Chap 4 (Variability)

Standard deviation

"Average" or typical deviation of the observations/scores around the mean; a measure of variability of data about the mean

Interquartile range (IQR)

Measure of variability recommended for skewed data or data with outliers IQR = Q3 - Q1

Range. When do you use the range as the measure of variability?

Nominal and ordinal data. " are used in what measure of variability?

When do you use the standard deviation?

ONLY when mean is the appropriate measure of center (the data is approximately symmetric with NO outliers).

μ

Population mean (meu)

Pro's and con's of standard deviation

Pro's: 1) Commonly used 2) Uses deviations from every data point 3) Has well-established theoretical properties Con's 1) Inflated by outliers and/or skewness

Σ( x - x̅ ) = ? Why is this NOT used in calculating the average of the deviations? What do you do instead?

0, because the positive deviations cancel out the negative deviations. The average of deviations will always = 0. Therefore, we want a number that is LIKE the average of deviations to know the average amount the scores differ from the mean (i.e. sample variance, standard deviation) Solutions: 1) SQUARE the deviations (i.e. variance) 2) SQUARE ROOT the variance (i.e. standard deviation)

μ = 80, 1) What is the mean? 2) What is the average amount that an individual scores deviates from the mean would be? 3) 68% of the population scores between what numbers?

1) 80 2) 6 3) 74 (80-6) and 86 (80+6)

Order these from small to large standard deviations. Which are more likely to have standard deviations of 5, 7, or 10?

1) B - Small, because most scores are around the mean (ex. standard deviation = 5). 2) C - Medium, because most scores are more spread out around mean (ex. standard deviation = 7) 3) A - Large, because most scores is the most spread out around the mean (ex. standard deviation = 10)

Why is the standard deviation the most commonly reported measure of variability in research?

1) It more directly communicates how consistently close the individual scores are to the mean. 2) It allows us to determine the middle 68% of the distribution.

What ways can you describe/determine variability?

1) Range (nominal, ordinal) 2) Variance 3) Standard Deviation

Descriptive measures used to describe a KNOWN: 1) sample: _________ or ________ 2) population: __________ or _________ 3) Final division by __________ Inferential measures are used to estimate the population based on a sample: 4) Final division by ____________ 5) Variance symbol _____________ 6) Standard deviation symbol _____________

1) S 2) S 3) N 4) N-1 5) s 6) s

Fill in the blank using: 1) When describing how far the scores are spread out from x̅, we use the __________ and ___________. 2) When describing how far the scores are spread out from μ, we use the __________ and ___________. 3) When the complete population of scores is unavailable, we infer the variability of the population based on a sample by computing the __________ (biased/unbiased) estimators __________ and ___________. These inferential formulas require a final _________________ instead of by _________.

1) S , S 2) , 3) Unbiased, s , s , division of N-1, N

What does variability communicate?

1) Small variability indicates that scores are consistently close to each other, and large variability indicates scores are inconsistent. 2) Amount of variability implies how accurately a measure of central tendency describes the distribution (i.e. the larger variability, the less accurately they are summarized by the mean, vice versa). 3) The distance/variability between scores can be seen as the distance that separates them (i.e. greater variability = greater distance between scores).

Why is sample variance not a good measure to use? How do you solve this problem?

1) Squaring deviations makes them unrealistically large. 2) It measures in square units (ex. if you're measuring ages, the variance indicates the scores deviate from the mean by 4 squared years.. whatever that means). Solution: Standard deviation - take a square root of the variance.

Order these from best to worst when measuring variability: 1) Standard deviation 2) Range 3) Variance 4) Sum of deviations

1) Standard deviation 2) Variance 3) Sum of deviations 4) Range

What are some guidelines to estimate the standard deviation?

1) Standard deviation is usually less than range/4 2) For large SYMMETRIC data sets, estimate the standard deviation by range/6 3) For large SKEWED data sets, estimate the standard deviation by range/5

You know there is an outlier when a score is ________ standard deviations from the mean.

3

If x̅ = 10 and S = 2, then 68% of the scores fall between __________ and _________.

8, 12

On an exam, the mean is 80 and the standard deviation is 5. What is the score at -1 S?

80 - 5 = 75

Sample variance

Average of the squared deviations of scores around the sample mean.

Population variance

Average squared deviation of scores around the population mean

What is the opposite of variability?

Consistency

How do you use the range with nominal data?

Counting the number of categories you're examining. Ex. There is more consistency if participants are part to 1 of 4 political parties, rather than 1 of 14 parties.

Variability

Describe the extent where scores vary from one another and the mean (aka. how spread out the distribution is)

The ______________ is the amount that a score deviates from the mean

Difference/variability

When the variability in a sample is large, are the scores close together or very different from each other?

Different

Range

Distance between highest and lowest scores in a set of data Range = Highest score - lowest score

Variability. " is also known as...

Distribution. " is also known as...

What is x - x̅ ?

Formula for the amount that a score deviates from the mean.

If a distribution is ____________, then the variability is large.

If a distribution is wide or spread out, then the variability is LARGE.

Why is the range not a good measure to determine variability?

It only uses the 2 most extreme scores. It is based on the least typical and least frequent scores, while ignoring all the other scores in the data set. This is why range is only used for NOMINAL and ORDINAL data.

In sample A, S = 6.82; Sample B, S = 11.41. Sample A is _______ (more/less) variable and most scores tend to be _________ (closer to/farther from) the mean.

Less, closer to

Pro's and con's of the range and IQR

Pro's: 1) Easy to compute 2) IQR is not sensitive to outliers Con's: 1) Doesn't show all data points 2) Neither is commonly used because of their complex theoretical properties 3) Range is sensitive to outliers

How do you use the range with ordinal data?

Range is the distance between the lowest and highest rank. Ex. If 100 runners finish a race with 5 positions (1st, 2nd, 3rd, 4th, 5th), it's a close race with many ties. BUT if they span 75 positions, the runners are more spread out.

Fill in the blank using: We use ____ and _____ to describe a sample, ____ and ____ to describe the true population, and _____ and ____ to estimate the population.

S , S , , , s , s

What is the difference between S and s ?

S is the sample standard deviation used to DESCRIBE a KNOWN population. s is the estimated population standard deviation used to INFER and ESTIMATE the population based on a sample.

Population standard deviation

Square root of the population variance (σ is called sigma)

Sample standard deviation

Square root of the sample variance

What unit does the variance use?

Squared Ex. If you are measuring ages, then the unit will be "squared" years.

What measure do you use to get the closest to the average of deviations as you can?

Standard deviation

When do you use the five-number summary vs. standard deviation?

Standard deviation is for SYMMETRIC data. Five-number summery is for SKEWED data.

What is the difference between computing the standard deviation and computing the variance?

Standard deviation is the square-root of the variance.

The LARGER the variance, the ______ the scores are spread out.

The __________ the variance, the MORE the scores are spread out.

Biased estimators

The formula for the variance or standard deviation involving a final division by N, used to describe a sample, but that tends to underestimate the population variability

How do the population standard deviation and populations variance differ?

The population standard deviation is a square root of the population variance

What is Σ( x - x̅ ) ?

The sum of squared deviations.

What does a large standard deviation describe about the typical opinion held by participants?

There is large disagreement among them.

What does a small standard deviation describe about participants exhibited a behavior?

They consistently exhibit the behavior.

When researchers measure the differences among scores, they measure...

Variability

What measures of variability are used for interval or ratio scores?

Variance and standard deviation

Which MEASURES OF VARIABILITY indicate how much the scores are spread out around the MEAN?

Variance and standard deviation. " are the two ______________ that indicate how much the scores spread out around the __________.

What unit does the standard deviation use?

Whatever unit you are measuring. Ex. If you are measuring ages, then the unit will be "years".

When scores are variable, we see frequent ________ differences among the scores, indicating that participants are behaving INCONSISTENTLY.

When scores are variable, we see frequent LARGE differences among the scores, indicating that participants are behaving _____________.

You know there is an outlier when a score is LESS than ________ IQR or GREATER than _____________ IQR .

You know there is an outlier when a score is _______ than Q1 - 1.5 IQR or ________ than Q3 + 1.5 IQR.

Symbol for the estimated population standard deviation

s

Symbol for the estimated population variance

s

Unbiased estimators

the formula for the variance or standard deviation involving a final division by N-1; calculated using sample data to estimate the population variability