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Basic Quantitative Methods in the Social

Sciences(AKA Intro Stats)

- 02-250-01
- Lecture 3

Variation

- Variability The extent numbers in a data set are

dissimilar (different) from each other - When all elements measured receive the same

scores (e.g., everyone in the data set is the

same age, in years), there is no variability in

the data set - As the scores in a data set become more

dissimilar, variability increases

Variation Range

- The range tells us the span over which the data

are distributed, and is only a very rough measure

of variability - Range The difference between the maximum and

minimum scores - Example The youngest student in a class is 19

and the oldest is 46. Therefore, the age range of

the class is 46 19 27 years.

Variation

- X
- 5 0.00 This is an example of data
- 5 0.00 with NO variability
- 5 0.00
- 5 0.00
- 5 0.00
- 25 n 5 5

Variation

- X
- 6 1.00 This is an example of data
- 4 -1.00 with low variability
- 6 1.00
- 5 0.00
- 4 -1.00
- 25 n 5 5

Variation

- X
- 8 3.00 This is an example of data
- 1 -4.00 with higher variability
- 9 4.00
- 5 0.00
- 2 -3.00
- 25 n 5 5

Note

- Lets say we wanted to figure out the average

deviation from the mean. Normally, we would want

to sum all deviations from the mean and then

divide by n, i.e., - (recall look at your formula for the mean from

last lecture) - BUT We have a problem. will

always add up to zero

Variation

- However, if we square each of the deviations from

the mean, we obtain a sum that is not equal to

zero - This is the basis for the measures of variance

and standard deviation, the two most common

measures of variability of data

Variation

- X

- 8 3.00

9.00 - 1 -4.00

16.00 - 9 4.00

16.00 - 5 0.00

0.00 - 2 -3.00

9.00 - 25 0.00

50.00 - Note The is called the Sum of

Squares

Variance of a Population

- VARIANCE OF A POPULATION the sum of squared

deviations from the mean divided by the number of

scores (sigma squared)

Population Standard Deviation

- Square root of the variance

Sample Variance

- the sum of squared deviations from the mean

divided by the number of degrees of freedom (an

estimate of the population variance, n-1)

Sample Standard Deviation

- Square root of the variance s2

Why use Standard Deviation and not Variance!??!

- Normally, you will only calculate variance in

order to calculate standard deviation, as

standard deviation is what we typically want - Why? Because standard deviation expresses

variability in the same units as the data - Example Standard deviation of ages in a class is

3.7 years

Variance

- The above formulae are definitional - they are

the mathematical representation of the concepts

of variance and standard deviation - When calculating variance and standard deviation

(especially when doing so by hand) the following

computational formulae are easiest to use (trust

us, they really are easier to use. You should

however have a good understanding of the

definitional formulae)

Population Variance

- Computational Formula

Population Standard Deviation

- Computational Formula

Sample Variance

- Computational Formula

Sample Standard Deviation

- Computational Formula

Sample Standard Deviation Example

- Data
- X X2
- 8 64 n 5, 5
- 1 1
- 9 81
- 5 25 s2 175 (25)2/5
- 2 4 4
- X25 175 s2 12.50
- s s 3.54

Computing Standard Deviation

- When calculating standard deviation, create a

table that looks like this

Computing Standard Deviation

- The values are then entered into the formula as

follows - 150

222 484 - n

4 -

n-1 3

Computing Standard Deviation

- The values are then entered into the formula as

follows - 150

222 484 - n

4 -

n-1 3

Computing Standard Deviation

- The values are then entered into the formula as

follows

Degrees of Freedom

- Degrees of Freedom The number of independent

observations, or, the number of observations that

are free to vary - In our data example above, there are 5 numbers

that total 25 ( 25, n 5)

Degrees of Freedom

- Many combinations of numbers can total 25, but

only the first 4 can be any value - The 5th number cannot vary if 25
- This example has 4 degrees of freedom, as four of

the five numbers are free to vary - Sample standard deviation usually underestimates

population standard deviation. Using n-1 in the

denominator corrects for this and gives us a

better estimate of the population standard

deviation.

Degrees of Freedom

- Degrees of freedom are usually
- n-1
- (the total of data points minus one)

Time for an example

- Seven people were asked to rate the taste of

McDonalds french fries on a scale of 1 to 10.

Their ratings are as follows - 8, 4, 6, 2, 5, 7, 7
- Calculate the population standard deviation
- Calculate the sample variance
- Class discussion When would this be a

population, and when would it be a sample?

Why is Standard Deviation so Important?

- What does the standard deviation really tell us?
- Why would a samples standard deviation be small?
- Why would a samples standard deviation be large?

An Example

- Youre sitting in the CAW Student Centre with 4

of your friends. A member of the opposite sex

walks by, and you and your friends rate this

persons attractiveness on a scale from 1 to 10

(where 1very unattractive and 10drop dead

gorgeous)

Food for thought

- 1) What would it mean if all five of you rated

this person a 9 on 10? - 2) What would it mean if all five of you rated

this person a 5 on 10? - 3) What would it mean if the five of you produced

the following ratings 1, 10, 2, 9, and 3 (note

that the mean rating would be 5)? - Why would scenario 3 happen instead of scenario

2? What factors would lead to these different

ratings? - These questions form the basis of why

statisticians like to explain variability

An In-Depth Look at Scenario 3

- So if the five of you produced the following

ratings 1, 10, 2, 9, and 3, what is the standard

deviation of these ratings? - Calculate!
- What is the standard deviation in Scenario 2?

Calculate!

Normal Distribution

- The normal distribution is a theoretical

distribution - Normal does not mean typical or average, it is

a technical term given to this mathematical

function - The normal distribution is unimodal and

symmetrical, and is often referred to as the Bell

Curve

Normal Distribution

Mean Median Mode

Normal Distribution

- We study the normal distribution because many

naturally occurring events yield a distribution

that approximates the normal distribution

Properties of Area Under the Normal Distribution

- One of the properties of the Normal Distribution

is the fixed area under the curve - If we split the distribution in half, 50 of the

scores of the sample lie to the left of the mean

(or median, or mode), and 50 of the scores lie

to the right of the mean (or median, or mode)

Properties of Area Under the Normal Distribution

- The mean, median, and mode always cut the Normal

Distribution in half, and are equal since the

Normal Distribution is unimodal and symmetrical

Properties of Area Under the Normal Distribution

50 of scores

50 of scores

Mean, Median, Mode

Properties of Area Under the Normal Distribution

- The entire area under the normal curve can be

considered to be a proportion of 1.0000 - Thus, half, or .5000 of the scores lie in the

bottom half (i.e., left of the mean) of the

distribution, and half, or .5000 of the scores

lie in the top half (i.e., right of the mean)

Properties of Area Under the Normal Distribution

.5000 of scores

.5000 of scores

Mean, Median, Mode

Z-scores

- Z-Scores (or standard scores) are a way of

expressing a raw scores place in a distribution

- Z-score formula

Z-scores

- The mean and standard deviation are

always notated in Greek letters - Z-scores only reflect the data points position

relative to the overall data set (so youre now

considering the data as a population, as youre

not looking to infer to a greater population) - This means use the population formula for

standard deviation rather than the sample formula

whenever you calculate Z

Z-scores

- A z-score is a better indicator of where your

score falls in a distribution than a raw score - A student could get a 75/100 on a test (75) and

consider this to be a very high score

Z-scores

- If the average of the class marks is 89 and the

(population) standard deviation is 5.2, then the

z-score for a mark of 75 would be - 89 5.2
- z (75-89)/5.2
- z (-14)/5.2
- z -2.69

Z-scores

- This means that a mark of 75 is actually 2.69

standard deviations BELOW the mean - The student would have done poorly on this test,

as compared to the rest of the class

Z-scores

- z 0 represents the mean score (which would be

89 in this example) - z lt 0 represents a score less than the mean

(which would be less than 89) - z gt 0 represents a score greater than the mean

(which would be greater than 89)

Z-scores

- For any set of scores
- the sum of z-scores will equal zero
- ( 0.00)
- have a mean equal to zero
- ( 0.00)
- and a standard deviation equal to one
- ( 1.00)

Z-scores

- A z-score expresses the position of the raw score

above or below the mean in standard deviation

sized units - E.g.,
- z 1.50 means that the raw score is 1 and

one-half standard deviations above the mean - z -2.00 means that the raw score is 2 standard

deviations below the mean

Z-score Example

- If you write two exams, in Math and English, and

get the following scores - Math 70 (class 55, 10)
- English 60 (class 50, 5)
- Which test mark represents the better performance

(relative to the class)?

Z-score Example cont.

- Math mark
- z (70-55)/10
- z 1.50
- English mark
- z (60-50)/5
- z 2.00

Z-score Example Illustration

Mean Z0.00

Z1.50

Z2.00

The Answer

- Because Z 2.00 is greater than Z 1.50, the

English class mark of 60 reflects a better

performance relative to that class than does the

Math class mark of 70

Z-score Solving for X

- The z-score formula can be rearranged to solve

for X

Z-scores Solving for X

- This formula is used when you know the z-score of

a data point, and want to solve for the raw score.

Example

- E.g., if a class midterm exam has 65 and

5, what exam mark has a z-score value of 1.25? - X (1.25)(5) 65
- 6.25 65
- 71.25
- So, a person whose test is 1.25 standard

deviations above the mean obtained a score of

71.25

Z-scores

- Z-score problems ask you to solve for X or solve

for z - Review both types of problems!

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