##### Measurement and Stats Ch. 3

Descriptive Statistics provides mathematical summaries of:

-Performance (best score)

-Performance Characteristics (central tendency and variability)

-Characteristics of distributions (symmetry)

3 scale levels of measurement:

-Nominal

-Ordinal

-Continuous (interval, ratio)

Nominal

-Categorical in nature

-identify differences based on a characteristic

-No notion of order, magnitude, or size

-Each category is as valuable as another is

-No arithmetic or comparisons

-Examples- gender, athletic position

Ordinal

-ranking

-items are ranked in order but do not indicate how much better one score is than the other

-Differences between ranked positions are not comparable

-Examples (finishing a race 1st 2nd 3rd, percentiles)

Continuous Interval

Numbers in which math operations can be performed and the results have meaning

-where the 0 point is chosen, 0 represents point on # line

-arithmetic can be done but ratios cannot be made

Ex: temp- 0 degrees does not mean temp doesnt exist

Interval

using an equal or common unit of measurement

Continuous Ratio

Ratio: # represent equal units between measurements & there is an absolute 0 point

-Ex: height, weight, distance

-Most often used in ES

Hierarchical

Builds on previous levels

Summation Notation

Mathematical Shorthand

Important Concepts:

n=# of subjects

x=measurement, observed variable

Sigma letter= sum of

Sigma letter (X) = x_{1 + }x_{2+ }X_{n}

Order of Operation

Parentheses- perform all operations within parentheses before moving outside them

If no parentheses, precedence rules of math

-conduct exponentiation

-follow with multiplication and division

-Follow with addition and subtraction

Summation Notation

EX^{2}-E^{X}/N X 2

Why and how do you report the data?

Why: want to summarize the info, tell people "how they did" (norm and criterion)

How: use a frequency distribution- method of organizing data that involves ranking scores and noting how often various scores occur

Breakdown of Step one to create a Frequency Distribution

1:Create column headings

-score

-tally

-frequency

-cumulative frequency

-percentage

-cumulative percentage (percentile)

Score

Interval, raw score

Tally

represents recording of scores (not always shown)

Frequency

(count) number of scores in the interval

Cumulative Frequency

cumulative frequency (count): numbers are obtained by summing the frequencies from bottom to top (should end up with N at the top)

Percentage Frequency

percentage of the frequency at a given interval; converts frequency to a percentage; FREQ/N

Cumulative Percentage

Cumulative percentage or percentile; converts cumulative frequencies to percentages; calculate by adding pct column from bottom to top or CUMFREQ?N* (preferred)

Steps 2-4 on how to create a frequency distribution

2: Find highest and lowest score then rank scores from high to low or best to worst

3:Determine number of times each score occurs-tally and count (frequency)

4:Determine cumulative frequency, percentages, and cumulative percentages

Percentiles

Represent ordinal scores; measures the position - there is no common unit of measurement between %tile ranks;quality of performance is arbitrary

Disadvantage - small changes in performance near the mean result in a disproportionate change in %tile rank; the opposite is true of extremes

Using Excel to calculate Percentiles

1: Prepare spreadsheet to present data in logical order - Type in appropriate %tiles

2: Place cursor where you want the answer to appear

3: click on fx, highlight percentile, OK

4: Determine array (a1:a20)

5:Type in which percentile you want to determine k(.2 = 20th percentile)

Using Excel to calculate Percentile Rank

1: Follow 1-2 above

2: Click on fx, highlight percentile rank, OK

3: Determine array and score/number (significance is not necessary)

Mode

-Most frequently observed score

-Unaffected by extreme scores

-Most easily estimated

-No calculations

-To obtain: Use frequency column and find greatest value

Median

-Exact middle of the distribution (ordered scores)

-50th percentile (ordinal)

-Not affected by extreme scores, more representative of central tendency with extreme scores

-Does not consider value, only rank

-To obtain: Order scores from high to low to find the middle one

-odd # of scores - middle

-Even # of scores - between 2 middle scores (add the middle scores then /2)

Mean

-Average

-Most sensitive of all measures of central tendency

-most appropriate for ratio data

-considers all information about the data

-influenced by extreme scores

-To obtain: sum of scores divided by the # of scores

Distribution shapes

Positively skewed: skewed to the right

Negatively skewed: Skewed to the left

Symmetrical: normal

Histogram

A graph that consists of columns to represent the frequencies with which various scores are observed

Vertical Axis

count (frequency)

Horizontal Axis

Scores

Range

Easiest measure of variability

Least stable measure of variability

To calculate: High - low scores

Variance

Reports heterogeneity of scores

Most stable measure of variability

2 sets of scores that have difference spreads will have different variances

Variance is also called mean square

Deviations are squared because it gives them desirable statistical properties

Variance is not in the original unit of measurement (SD)

True Variance

Scores are "really difference"

Error Variance

Scores are difference because of errors made

Standard Deviation

Square root of the variance

illustrates the variability of a set of scores

linear measurement

presented in the same units as the scores (PPF)

determine heterogeneity or homogeneity of scores

Standard Scores

a set of observations that have been standardized around a given mean and standard deviation

used to compare scores of different units (1 mile run, sit ups)

informs participants of how they did on test compared to others

determines percentiles

Z score

subtract mean from the observed and divide by s

provides a logical means of comparison

difficult to explain negative #

indicates the # of SD that a score is above or below the mean

Normal curve areas

Purpose: to determine percentiles and % of observations that fall within a particular area under the curve

The area under the normal curve is 100%; the area represented in the chart is 50%

relationship of measures of central tendency in normal distribution

mean is 50th percentile (mean = median = mode