Chapter 1

The mean, the number of observations, the variance and the standard deviation

Some definitions

Data - observations, measurements, scores

Statistics - a series of rules and methods that can be used to organize and interpret data.

Descriptive Statistics - methods to summarize large amounts of data with just a few numbers.

Inferential Statistics - mathematical procedures to make statements of a population based on a sample.

More DefinitionsParameter - a number that summarizes or

describes some aspect of a population.

Sample statistic - An estimate of a population parameter based on a random sample taken from the population.

Sampling Error - the difference between a sample statistic that estimates a population parameter and the actual parameter.

Non-parametric Statistics - statistics for observations that do not allow the estimation of the population mean and variance.

Sampling Error - the difference between a sample statistic that estimates a population parameter and the actual parameter.

Differences between sample statistics and population parameters are largely a function of stable, random individual differences and measurement problems

Where we are going

Descriptive Statistics

Number of Observations

Measures of Central Tendency

Measures of Variability

Observations

Each score is represented by the letter X.

The total number of observations is represented by N.

Measures of Central Tendency

Finding the most typical score

median - the middle score

mode - the most frequent score

mean - the average score

In this course, the mean will be

our most important measure of

central tendency

Greek letters are used to represent population parameters.

(mu) is the mathematical symbol for the mean.

is the mathematical symbol for summation.

Formula - = (X) / NEnglish: To calculate the mean, first add

up all the scores, then divide by the number of scores you added up.

Calculating the Mean

The mode, the median and the mean

606345646570556066

Ages of people retiring from Rutgers this year. 455560606364656670

X = 548N = 9

Mean = 60.89

Mode is 60. Median is 63.

Measures of Variability – less important

Range - the distance from the highest to the lowest score.

Inter-quartile Range - the distance from the top 25% to the bottom 25%.

Sum of Squares (SS) – the total squared distance of all scores from the mean. You calculate it by finding the distance of each score from the mean, squared and then summed over all the scores.

Measures of Variability – more important

Variance (2)- also called sigma2. The variance is the average squared distance of scores from mu. It is found by dividing the total squared distance of all the scores from the mean and then dividing by the

number of scores (2=SS/N)

Standard Deviation ()- also called sigma. The standard deviation is the square root of the variance. It is the average unsquared distance of scores in the population from their mean. (That is almost, but not exactly like saying that the standard deviation is the average distance of scores from the population mean.)

Computing the variance and the standard deviation

Scores on a 10 question Psychology quiz

Student

John

JenniferArthurPatrickMarie

X

7

8357

X = 30 N = 5 = 6.00

X -

+1.00

+2.00-3.00-1.00+1.00

(X- ) = 0.00

(X - )2

1.00

4.009.001.001.00

(X- )2 = SS = 16.00

2 = SS/N = 3.20 = = 1.7920.3

The variance is our most basic and important measure of variability

The variance ( =sigma squared) is the average squared distance of individual scores from the population mean.

Other indices of variation are derived from the variance.

For example,. as noted above, sigma is the average unsquared distance of scores from mu is the standard deviation. To find it you compute the square root of the variance.

2

Other measures of variability derived from the variance We can randomly choose scores from a population

to form a random sample and then find the mean of such samples.

Each score you add to a sample tends to correct the sample mean back toward the population mean, mu.

The average squared distance of sample means from the population mean is the variance divided by n, the size of the sample.

To find the average unsquared distance of sample means from mu divide the variance by n, then take the square root. The result is called the standard error of the sample mean or, more briefly, the standard error of the mean. We’ll see more of this in Ch. 4.

Making predictions (1)Without any other information, the

population mean (mu) is the best prediction of each and every person’s score.

So you should predict that everyone will score precisely at the population mean.

Why? Because the mean is an unbiased predictor or estimate. The mean is as close to the high as to the low scores in the population.

This is mathematically proven by the fact that deviations around the mean sum to zero.

You should also predict that everyone will score right at the mean because:

The mean is the number that is the smallest average squared distance from all the scores in the distribution.

Thus, the mean is your best prediction, because it is a least squares, unbiased predictor.

What happens if we make a prediction other than mu.Scores on a Psychology quiz (mu = 6.00) What if we predict everyone will score 5.50? Deviations don’t sum to zero and the average squared distance of scores from the prediction increases

Student

John

JenniferArthurPatrickMarie

X

7

8357

X = 30 N = 5 = 6.00

X - 5.5

+1.50

+2.50-2.50-0.50+1.50

(X- ?) = 2.50

(X - )2

2.25

6.256.250.252.25

(X- ?)2 = SS = 17.25

2 = SS/N = 3.45 = = 1.8620.3

X - 5.50 (X - 5.50)2

Compare that to predicting that everyone will score right at the mean (mu).

Scores on a 10 question Psychology quiz

Student

John

JenniferArthurPatrickMarie

X

7

8357

X = 30 N = 5 = 6.00

X -

+1.00

+2.00-3.00-1.00+1.00

(X- ) = 0.00

(X - )2

1.00

4.009.001.001.00

(X- )2 = SS = 16.00

2 = SS/N = 3.20 = = 1.7920.3

But when you predict that everyone will score at the mean, you will be wrong. In fact, it is often the case that no one will score precisely at the mean.

In statistics, we don’t expect our predictions to be precisely right.

We want to make predictions that are wrong in a particular way.

We want our predictions to be as close to the high scores as to the low scores in the population.

The mean is the only number that is an unbiased predictor, it is the only number around which deviations sum to zero.

We want to be wrong by the least amount possible In statistics, we consider error to be the

squared distance between a prediction and the actual score.

The mean is the least average squared distance from all the scores in the population.

The number that is the least average squared distance from the scores in the population is the prediction that is least wrong, the least in error.

Thus, saying that everyone will score at the mean (even if no one does!) is the prediction that gives you the smallest amount of error.

Why doesn’t everyone score right at the mean?Sources of Error

Individual differences – people have stable differences from one another. They differ in an infinite number of ways and combination of ways.

PROOF OF THAT: AREN’T YOU ARE MORE LIKE WHO YOU WILL BE IN 5 MINUTES THAN YOU ARE LIKE THE PERSON NEXT TO YOU??!

AND – THERE ARE ALWAYS MEASUREMENT PROBLEMS! Instruments are imperfect, scores get mistranscribed, participants may be uninterested or have a stomach ache, etc. etc. etc. …

Remember: THERE ARE ALWAYS MEASUREMENT PROBLEMS

NO MEASUREMENT DEVICE IS EVER PERFECTLY ACCURATE, WHETHER IT IS A HIGHLY ACCURATE SCALE OR A 12 QUESTION QUESTIONNAIRE

Additionally, transient situational factors make measurement inaccurate

This is especially true when we measure people. Let’s say we are measuring something relatively easy to measure, such as verbal ability. When we are measuring people, lots of transient factors (such as mood, events, time, motivation etc.) all change an individual’s responses and combine to make our measurement of verbal ability imperfect.

The mean square for error

We call the average squared error of prediction when we use the mean as our prediction the “mean square for error”. It tells us how much (squared) error we make, on the average, when we predict that everyone will score precisely at the mean.

Mean square for error = the variance (sigma2)

If we predict that everyone will score right at the mean, how much error do you make on the average? To find out, find the distance of each score from the mean, square that distance and divide by the number of scores to find the average error.

WHOOPS: THAT’S SIGMA2.

Questions and answers – the mean.

WHAT QUALITIES OF THE MEAN (MU) MAKE IT THE BEST PREDICTION YOU CAN MAKE OF WHERE EVERYONE WILL SCORE?

The mean is an unbiased predictor or estimate, because the deviations around the mean always sum to zero.

The mean is a least squares predictor because it is the smallest squared distance on the average from all the scores in the population.

So the variance has a third name.

The variance is called the mean square for error as well as being called sigma2.

As the mean square for error, the variance is our numerical index of the effects of individual differences and measurement problems.

Q & A: the mean

WHY WOULD YOU PREDICT THAT EVERYONE WILL SCORE AT THE MEAN WHEN, IN FACT, OFTEN NO ONE CAN POSSIBLY SCORE PRECISELY AT THE MEAN?

In statistics, we don’t expect our predictions to be precisely right.

We want to make predictions that are close and wrong in a particular way.

We want least squares, unbiased predictors.

Q & A: The variance

WHAT ARE THE OTHER NAMES FOR THE VARIANCE?

Sigma2 and the mean square for error.WHAT OTHER MEASURES OF

VARIABILITY CAN BE EASILY COMPUTED ONCE YOU KNOW THE VARIANCE?

The standard deviation and the standard error of the sample mean.

How do you compute

THE VARIANCE? Find the distance of each score from the mean, square it, sum them up and divide by the number of scores in the population.

THE STANDARD DEVIATION? Compute the square root of the variance.

THE STANDARD ERROR OF THE SAMPLE MEAN? Divide the variance by n, the size of the sample, and then take a square root.

END CHAPTER 1 SLIDES