outline i.what are z-scores? ii.locating scores in a distribution a.computing a z-score from a raw...

Outline

I. What are z-scores?

II.Locating scores in a distributionA. Computing a z-score from a

raw score

B. Computing a raw score from a z-score

C. Using z-scores to standardize distributions

III. Comparing scores from different distributions

I. What are z scores?

You scored 76How well did you perform? serves as reference point: Are you above or below average?

serves as yardstick:How much are you above or below?

• Convert raw score to a z-score• z-score describes a score relative to & • Two useful purposes:• Tell exact location of score in a distribution• Compare scores across different

distributions

II. Locating Scores in a distribution

Deviation from in SD units

Relative status, location, of a raw score (X)

z-score has 2 parts:

1. Sign tells you above (+) or below (-)

2. Value tells magnitude of distance in SD units

Xz

A. Converting a raw score (X) to a z-score:

Example:

Spelling bee: = 8 = 2

Garth X=6 z =

Peggy X=11 z =

Xz

B. Converting a z-score to a raw score:

Example:

Spelling bee: = 8 = 2

Hellen z = .5 X =

Andy z = 0 X =

raw score = mean + deviation

zX

C. Using z-scores to Standardize a Distribution

Convert each raw score to a z-scoreWhat is the shape of the new dist’n?

Same as it was before! Does NOT alter shape of dist’n!

Re-labeling values, but order stays the same!

What is the mean? = 0Convenient reference point!

What is the standard deviation? = 1

z always tells you # of SD units from !

An entire population of scores is transformed into z-scores. The transformation does not change the shape of the population but the mean is transformed into a value of 0 and the standard deviation is transformed to a value of 1.

Example:

So, a distribution of z-scores always has:

= 0 = 1A standardized distribution

helps us compare scores from different distributions

Student X X- zGarth 6

Peggy 11

Andy 8

Hellen 9

Humphry 5

Vivian 9

N = 6 N=6

= 8 = 0

= 2 = 1

III. Comparing Scores From Different Dist’s

Example:

Jim in class A scored 18

Mary in class B scored 75

Who performed better?Need a “common metric”

Express each score relative to it’s own &

Transform raw scores to z-scores

Standardize the distributions

they will now have same &

Example:

Class A: Jim scored 18 = 10 = 5

Class B: Mary scored 75 = 50 = 25

Who performed better? Jim!

Two z-scores can always be compared

6.151018 z

125

5075 z

Outline: Probability and The Normal Curve

I. ProbabilityA. Probability and inferential statistics

B. What is probability?

II. The Normal CurveA. Probability and the Normal Curve

B. Properties of the Standard Normal Curve

C. The Unit Normal Table

III. Solving Problems with the Normal Curve

A. Problem Type 1

B. Problem Type 2

C. Cautions

I. ProbabilityA. Probability & Inferential Statistics

Transition to inferential statisticsWhy is probability so important? Links samples and populations

Example 1:The jar is a “population”One marble is a “sample” How likely to get BLACK?But, isn’t the goal of inferential stats the

opposite? Example 2:

Choose 10 marbles, blindfolded “Sample” has 8 BLACK & 2 WHITEWhich jar did marbles come from?

This is inferential statistics! “Judgments under uncertainty”

B. What is probability?Likelihood of an “event” occurring

Can range from 0 (never) to 1.0 (always)

Defined in terms of a fraction, proportion, or percentage

p(A) = Number of outcomes classified as A Total number of possible outcomes

Example #1:Toss a coin, what is probability of heads?

p(Heads) = 1/21 = one way to get heads2 = two possible outcomes (heads or tails)

½ = .50 = 50%

Example #2:

Select a card from a deck of 52 cardsWhat is probability of selecting a king?

p(King) = 4/52

4 = four ways to get a king

52 = 52 possible outcomes

4/52 = .077 = 7.7%

Compute probability from a frequency distribution

What is the probability of selecting a score with x = 8?

p(x = 8) = f/N = 3/10 = .30 = 30%

What is the probability of selecting a score with x < 8?

p(x < 8) = f/N = 6/10 = .60 = 60%

Nfp

X f

9 1

8 3

7 4

6 2

Σf = N = 10

II. The Normal Curve

A. Probability and the Normal Curve

– Special statistical tool called the Normal Curve

– Theoretical curve defined by mathematical formula

– Known proportions/areas under the curve

– Used to solve problems when we don’t know the population

B. Properties of the Standard Normal Curve

• Theoretical, idealized curve• Based on mathematical formula• Bell-shaped, symmetrical, unimodal• μ = Md = Mo• 50% of scores above m, 50% below• Standardized: μ = 0, σ = 1 • A probability distribution, tails not anchored to axis• Total area under the curve will sum to 1.0• Exact percentiles associated with each z-score• Area under curve provided in Unit Normal Table• Can be applied to any normal distribution once the

distribution is standardized (converted to z-scores)

Why is the normal curve so important?

(1) Many variables normally distributed in population

(2) Can use normal curve to solve many problems

Two types of problems:(1) What proportion of dist’n falls

above, below, or between particular z-scores?

(2) What z-score is associated with particular proportions/probabilities under the curve?

C. The Unit Normal Table (UNT)

A = z-scores

B = Proportion in body (larger portion)

C = Proportion in tail (smaller portion)

• Curve is symmetrical, only + z scores shown

• Columns B & C always sum to 1.0

• Proportions/probabilities are always positive

z-score cuts curve into two portions (B & C)

Let’s Practice!Tip: Always sketch a

curve first!

Examples 1:

What proportion of distribution falls above z = 1.5?

p (z > 1.5)

What proportion falls below z = -.5?

p (z < -0.5)

Examples 2:

What z-score separates the lower 75% from the upper 25%? (same as 75th percentile)

What z-scores separate the middle 60% of the distribution from the rest of the distribution?

II. Solving Problems with the Normal Curve

Hints and TipsTwo types of problems: (1) Finding proportion associated with X or z

(2) Finding X or z associated with proportion

Problem Type #1 Steps to Follow: (a) Sketch curve(b) Convert raw score to z-score(c) Look up proportion for this z-score (d) Sometimes add/subtract proportions

Problem type #2 Steps to Follow:(a) Sketch curve(b) Look up z-score associated with proportion(c) Convert z-score back to a raw score (X)

Always sketch a normal curve first!

A. Problem Type 1: Finding Area Under the Curve

Problem #1:

Exam: μ = 60 σ = 10

What percentage will score below 70? (1) Sketch a normal curve

(2) Convert raw score to z-scorez =

(3) Plan your strategy

(4) Refer to Unit Normal Table

Problem #2:

Exam μ = 60 σ = 10

What is percentile rank of student who scored 55?

(1) Sketch a normal curve

(2) Convert raw score to z-score

z =

(3) Plan your strategy

(4) Refer to UNT

Problem #3:

Exam μ = 60 σ = 10What proportion of people will

score between 60 and 80?(1) Sketch a normal curve

(2) Convert raw score to z-score

z =

(3) Plan your strategy

(4) Refer to UNT

Problem #4:

Exam: μ = 60 σ = 10

What proportion of people will score between 50 and 80?

(1) Sketch a normal curve

(2) Convert raw scores to z-scores

z1 =

z2 =

(3) Plan your strategy

(4) Refer to UNT

B. Problem Type 2: Finding a Score Associated

with a Proportion or Percentile

Problem #5: Standardized Exam: μ = 60 σ = 10

Assign A+ to the 95th percentile What is cut-off score for earning an

A+? (1) Sketch curve(2) Plan your solution(3) Refer to UNT

z = (4) Convert z-score back to raw

score:

x = + z σx =

Problem #6:

Exam: μ = 60 σ = 10Assign F to 15th percentile (and

below) What is cut-off score for earning an

F? (1) Sketch curve(2) Plan your solution(3) Refer to UNT z = (4) Convert z-score back to raw

score:

x = + z σx =

C. Cautions In order to use the UNT to solve problems,

you must:

• have known μ and σ• assume your variable is normally distributed

Why?

• If you don’t know μ & σ, can’t compute a z-score

• If variable is not normally distributed, percentages given by UNT won’t apply!

• z-scores can be negative but proportions/ percentiles cannot!

• Pay close attention to the words…

– Above, Below, Within, Beyond

The Distribution of Sample Means

Inferential statistics:Generalize from a sample to a population

Statistics vs. ParametersWhy?

Population not often possibleLimitation:

Sample won’t precisely reflect populationSamples from same population vary

“sampling variability”Sampling error = discrepancy between sample statistic and population parameter

The Distribution of Sample Means

• Extend z-scores and normal curve to SAMPLE MEANS rather than individual scores

• How well will a sample describe a population?

• What is probability of selecting a sample that has a certain mean?

• Sample size will be critical– Larger samples are more

representative– Larger samples = smaller error

The Distribution of Sample Means

Population of 4 scores: 2 4 6 8 = 5

4 random samples (n = 2):

is rarely exactly

Most a little bigger or smaller than

Most will cluster around

Extreme low or high values of are relatively rare

With larger n, s will cluster closer to µ (the DSM will have smaller error, smaller variance)

We don’t actually compute a DSM!

62X 53X 34X 41X

X

X

X

X

X

A Distribution of Sample Means

The distribution of sample means for n = 2. This distribution shows the 16 sample means obtained by taking all possible random samples of size n=2 that can be drawn from the population of 4 scores. The known population mean from which these samples were drawn is µ = 5.

X=4 X=5 X=6

The Distribution of Sample Means

A distribution of sample means ( )

All possible random samples of size n

A distribution of a statistic (not raw scores)

“Sampling Distribution” of

Probability of getting an , given known and

Important properties(1) Mean(2) Standard Deviation(3) Shape

X

X

X

Properties of the DSM

Mean?

Called expected value of

Standard Deviation?

Any can be viewed as a deviation from

= Standard Error of the Mean

Variability of around

Special type of standard deviation, type of “error”

Average amount by which deviates from

x

X

uuX

X

X

X

nσ /x

Less error = better, more reliable, estimate of population parameter

influenced by two things:

(1) Sample size (n)Larger n = smaller standard errorsNote: when n = 1

as “starting point” for gets smaller as n increases

(2) Variability in population ()

Larger = larger standard errors

Note:

x

mx

xx x

The distribution of sample means for random samples of size (a) n = 1, (b) n = 4, and (c) n = 100 obtained from a normal population with µ = 80 and σ = 20. Notice that the size of the standard error decreases as the sample size increases.

Shape of the DSM?

Central Limit Theorem = DSM will approach a normal dist’n as n approaches infinity

Very important! True even when raw scores NOT normal! True regardless of or

What about sample size? (1) If raw scores ARE normal, any n will do (2)If raw scores NOT normal, n must be “sufficiently large”

For most distributions n 30

Why are Sampling Distributions important?• Tells us probability of getting , given &

• Distribution of a STATISTIC rather than raw scores

• Theoretical probability distribution

• Critical for inferential statistics!

• Allows us to estimate likelihood of making an error when generalizing from sample to popl’n

• Standard error = variability due to chance

• Allows us to estimate population parameters

• Allows us to compare differences between sample means – due to chance or to experimental treatment?

• Sampling distribution is the most fundamental concept underlying all statistical tests

X

Working with the Distribution of Sample

Means• If we assume DSM is normal • If we know & • We can use Normal Curve & Unit

Normal Table!

Example #1:

= 80 = 12

What is probability of getting 86

if n = 9?

x

Xz

X

Example #1b:

= 80 = 12

What if we change n =36

What is probability of getting 86 X

Example #2:

= 80 = 12

What marks the point beyond which sample means are likely to occur only 5% of the time? (n = 9)

X