PSY 307 – Statistics for the

Behavioral Sciences

Chapter 3-5 – Mean, Variance, Standard Deviation and Z-scores

Measures of Central Tendency

(Representative Values)

Quantitative data:

Mode – the most frequently occurring observation

Median – the middle value in the data

Mean – average

Qualitative data:

Mode – can always be used

Median – can sometimes be used

Mean – can never be used

Mode

The value of the most frequently occurring observation.

In a frequency distribution, look for the highest frequency.

In a graph, look for the peaks or highest bar in a histogram.

Distributions with two peaks are bimodal (have two modes). Even if the peaks are not exactly the

same height.

Median

The middle value when observations are ordered from least to most, or vice versa.

Half the numbers are higher and half are lower.

When there is an even number of observations, the median is the average of the two middle values.

Mean

The most commonly used and most useful average.

Mean = sum of all observations number of all observations = X

n

Observations can be added in any order.

Notation

Sample vs population

Population notation = Greek letters

Individual value = x (lower case)

Sample mean = x or M

Population mean =

Summation sign =

Sample size = n

Population size = N

Mean as Balance Point

The sum of the deviations from the mean always equals zero.

The mean is the single point of equilibrium (balance) in a data set.

The mean is affected by all values in the data set.

If you change a single value, the mean changes.

Demo

The Most Descriptive Average

When a distribution is not skewed (lopsided), the mean, median & mode are similar.

When a distribution is skewed, the mean is closer to the extreme values, mode is farthest. Report both the mean and median for a

skewed distribution.

The mean is the preferred average.

Ranked Data

Mean and modal ranks are not informative.

The mean always equals the median (middle) rank, so use the median.

The mode occurs when there is a tie in the data, but doesn’t mean much.

Find the median by finding the middle rank (or the average of the two middle ranks).

Qualitative Data Averages

The mode can always be used.

The median can only be used when classes can be ordered. The median is the category that

contains 50% in its cumulative frequency.

Never report a median with unordered classes.

Never report the mean.

Measures of Variability

Range – difference between highest and lowest value.

Variance – the mean of the squared deviations (differences) from the mean.

Standard Deviation – square root of the variance. The average amount that observations

deviate from the mean.

Interquartile Range (IQR)

The range for the middle 50% of observations. Distance between the 25th and 75th

percentiles.

Remove the highest and lowest 25% of scores then calculate the range for the remaining values.

Used because it is insensitive to extreme observations.

Using IQR (from Holcomb)

In Rio, what percentage had been injecting from 4.5 to 14 years?

Median Year Injecting = 10

IQR is 4.5-14 (from text).

0 100%

Median = 50%

IQR

4.5 14 25% 25% 25% 25%

More Notation

Sample variance = S2

Population variance = 2

Sample standard deviation = S or SD

Population standard deviation =

Interquartile range = IQR

What Does Variance Describe?

Variance and standard deviation describe the amount that actual observations differ from the mean. How spread out are the scores?

The range doesn’t tell us how scores are distributed between the high and low values.

Because the mean is the balance point, the mean of the unsquared deviations is always zero.

An example using dogs.

Mean = 600 + 470 + 170 + 430 + 300 = 1970 = 394 mm

5 5

Source of example using dogs: http://www.mathsisfun.com/standard-deviation.html

First calculate the height of the dogs.

Next, compare their heights to the

mean.

The green line shows the mean. Subtract the mean from

each dog’s height. Because some dogs are taller and

others are shorter, some of the differences will be positive

and some negative numbers. These differences will

cancel each other out because the mean is the balance

point in the distribution of dog heights.

Square the differences and take

the mean.

σ2 = 2062 + 762 + (-224)2 + 362 + (-94)2 = 108,520 = 21,704

5 5

Take the square root to return to

the original units of measure.

σ = √21,704 = 147

Which dogs are within one standard deviation of the mean?

Rottweillers are unusally tall dogs. And Dachsunds

are a bit short.

Standard Deviation

The variance is expressed in squared units (e.g., squared lbs) which are hard to interpret.

Taking the square root of the variance expresses the average deviation in the original units.

The square root of the variance gives a slightly different result than taking the average of the absolute deviations.

Interpreting the SD

For most distributions, the majority of observations fall within one standard deviation of the mean.

A very small minority fall outside two standard deviations.

This generalization is true no matter what the shape of the distribution.

It works for skewed distributions.

A Measure of Distance

The mean shows the position of the balance point within a distribution.

The standard deviation is a unit of distance that is useful for comparing scores.

Standard deviations cannot have a negative value. They can measure in both positive and

negative directions from the mean.

Definition Formula

Definition formula – easier to understand conceptually.

The numerator is also called the Sum of the Squares (squared differences), abbreviated SS

N

XX 2)(2)( XX

Computation Formula

Computation formula – easier to use, especially with large data sets.

The computational and definition formulas produce the same result.

n

XX 22

2)(

N

SS

Population vs Sample

The formulas are different depending on whether a sample or a population is being measured.

Use n-1 in the denominator when using s or s2 to estimate or 2 for a population.

Using n-1 more accurately estimates the variability in a population.

Formulas

Variance for sample:

Variance for population:

1

2

n

SSs

N

SS2

Z-Score

Indicates how many SDs an observation is above or below the mean of the normal distribution.

Formula for converting any score to a z-score:

Z = X –

mean std. deviation

Properties of z-Scores

A z-score expresses a specific value in terms of the standard deviation of the distribution it is drawn from.

The z-score no longer has units of measure (lbs, inches).

Z-scores can be negative or positive, indicating whether the score is above or below the mean.

Standard Normal Curve

By definition has a mean of 0 and an SD of 1.

Standard normal table gives proportions for z-scores using the standard normal curve.

Proportions on either side of the mean equal .50 (50%) and both sides add up to 1.00 (100%).

Other Distributions

Any distribution can be converted to z-scores, giving it a mean of 0 and a standard deviation of 1.

The distribution keeps its original shape, even though the scores are now z-scores. A skewed distribution stays skewed.

The standard normal table cannot be used to find its proportions.

Transformed Standard Scores

Z-scores are useful for converting between different types of standard scores:

IQ test scores, T scores, GRE scores

The z-scores are transformed into the standard scores corresponding to standard deviations (z).

New score = mean + (z)(std dev)