Statistics

Review of Statistics

Levels of Measurement

Descriptive and Inferential Statistics

Levels of MeasurementNature of the variable affects rules applied to its measurement

Qualitative Data Nominal Ordinal

Quantitative Data Interval Ratio

Nominal Measurement

Lowest Level Sorting into categories Numbers merely symbols--have no

quantitative significance Assign equivalence or nonequivalence

Examples, gender, marital status, etc

Male / female smoker /nonsmoker alive/dead

1 2

Rules of Nominal system

All of members of one category are assigned same numbers

No two categories are assigned the same number (mutual exclusivity)

Cannot treat the numbers mathematically

Mode is the only measure of central tendency

The Ordinal Scale

Sorting variations on the basis of their relative standing to each other

Attributes ordered according to some criterion (e.g. best to worst)

Intervals are not necessarily equal

Should not treat mathematically, frequencies and modes ok

Ordinal scale

0 1 2 3 4

Interval Scale

Researcher can specify rank ordering of variables and distance between

Intervals are equal but no rational zero point (example IQ scale, Fahrenheit scale)

Data can be treated mathematically, most statistical tests are possible

Ratio Scale

Highest level of measurement Rational meaningful zero point Absolute magnitude of variable (e.g.,

mgm/ml of glucose in urine) Ideal for all statistical tests

Descriptive Statistics

Used to describe data Frequency distributions, histograms,

polygons Measures of Central Tendency Dispersion Position within a sample

Frequency Distributions

Imposing some order on a mass of numerical data by a systematic arrangement of numerical values from lowest to highest with a count of the number of times each value was obtained--Most frequently represented as a frequency polygon

Frequency distribution

0

5

10

15

20

25

30F

req

ue

ncy

Shapes of distributions

Symmetry Modality Kurtosis

Symmetry

Normal curve symmetrical If non symmetrical skewed (peak is off

center)– positively skewed– negatively skewed

Positive skew

Negative skew

Modality

Describes how many peaks are in the distribution– unimodal– bimodal– multimodal

unimodal

bimodal

multimodal

Kurtosis

Peakedness of distribution– platykurtic – mesokurtic– leptokurtic

Mesokurtic

Platykurtic

Leptokurtic

Measures of Central Tendency

Overall summary of a group’s characteristics

“What is the average level of pain described by post hysterectomy pts.?”

“How much information does the typical teen have about STDs?”

Mean

Arithmetic average Most widely reported meas. of CT Not trustworthy on skewed distributions

Median

The point on a distribution above which 50% of observations fall

Shows how central the mean really is since the median is the number which divides the sample in half

Does not take into account the quantitative values of individual scores

Preferred in a skewed distribution

Mode The most frequently occuring score or

number value within a distribution Not affected by extreme values Shows where scores cluster There may be more than one mode in a

distribution Arrived at through inspection limited usefulness in computations

Which measures of central tendency is represented by each of these lines?

Variability or Dispersion Measures

Percentile rank-the point below which a % of scores occur

Range --highest-lowest score Standard deviation--master measure of

variability--average difference of scores from the mean--allows one to interpret a score as it relates to others in the distribution

Normal (Gaussian) Distribution

Mathematical ideal– 68.3% of scores within +/- 1sd– 95.4% of scores within +/- 2sd– 99.7% of scores within +/- 3sd

unimodal

mesokurtic

symmetrical

Normal curve

1% 13.5% 34% 34% 13.5 % 1 %

Inferential Statistics

Used to make inferences about entire population from data collected from a sample

Two classifications based on their underlying assumptions

Parametric Nonparametric

Parametric Based on population parameters Have numbers of assumptions

(requirements) Level of measurement must be interval

or ratio– t-test– Pearson product moment correlation ®– ANOVA– Multiple regression analysis

Parametric

Preferable because they are more powerful--better able to detect a significant result if one exists.

Nonparametric

Not as powerful Have fewer assumptions Level of measurement is nominal or

ordinal– Chi squared

Some examples of Statistical tests and their useStatistical Test Purpose IV DV

t-test (t) To test the differencebetween 2 gp. means

nominal Interval or ratio

ANOVA (F) To test the differenceof means among 3ormore gps

Nominal Interval or ratio

Pear. ProdMom. Corr (r )

To test that arelationship exists

Interval orordinal

Interval orordinal

Chi Squaredtest (X2)

To test the differencesin proportions in 2 ormore groups todetermin if results arepossible due tochance

Nominal Nominal

analysed with: Analyse-It + General v1.40 

Test  Chi-square test Caffeine consumption of adults Marital status by Caffeine consumption

Performed by  Analyse-it Software, Ltd. Date 1 February 1999

n 3888  

Count Caffeine consumptionMarital status 0 1-150 151-300 >300 Total

Married  652 1537 598 242 3029  (705.8) (1488.0) (578.1) (257.1)  

Divorced, seperated, widowed  36 46 38 21 141  (32.9) (69.3) (26.9) (12.0)  

Single  218 327 106 67 718  (167.3) (352.7) (137.0) (60.9)  

Total  906 1910 742 330 3888

X² statistic 51.66p  <0.0001  

Hypothesis testing

Research Hypothesis Hr--Statement of the researcher’s prediction

Alternate Hypothesis Ha--Competing explanation of results

Null Hypothesis Ho -- Negative Statement of hypothesis tested by statistical tests

Research Hypotheses

Method A is more effective than method B in reducing pain (directional)

Method A will differ from Method B in pain reducing effectiveness (nondirectional)

Null Hypothesis

Method A equals Method B in pain reduction effectiveness.(any difference is due to chance alone

This must be statistically tested to say that something else beside chance is creating any difference in results

Type I and Type II errors

Type I--a decision to reject the null hypothesis when it is true. A researcher conludes that a relationship exists when it does not.

Type II--a decisioon to accept the null hypothesis when it is false. The researcher concludes no relationship exists when it does.

Level of Significance

Degree of risk of making a Type one error. (saying a treatment works when it doesn’t or that a relationship exists when there is none)

Signifies the probability that the results are due to chance alone.

p=.05 means that the probability of the results being due to chance are 5%