Statistics

Review of Statistics Levels of Measurement Descriptive and Inferential Statistics

Levels of Measurement Nature 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 10 5 0 30 25 20 15

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 use

Statistical Test t-test (t) ANOVA (F) Pear. Prod Mom. Corr (r ) Chi Squared test (X 2 ) Purpose IV To test the difference between 2 gp. means nominal To test the difference of means among 3or more gps To test that a relationship exists Nominal Interval or ordinal To test the differences in proportions in 2 or more groups to determin if results are possible due to chance Nominal DV Interval or ratio Interval or ratio Interval or ordinal Nominal

Test Chi-square test

Caffeine consumption of adults Marital status by Caffeine consumption

Performed by

Analyse-it Software, Ltd.

n

3888

Count Marital status

Married Divorced, seperated, widowed Single

Total

0 652 (705.8) 36 (32.9) 218 (167.3)

906 Caffeine consumption

1-150 151-300 1537 (1488.0) 46 (69.3) 327 (352.7)

1910

598 (578.1) 38 (26.9) 106 (137.0)

742

>300 242 (257.1) 21 (12.0) 67 (60.9)

330 Total 3029 141 718 3888 X² statistic p

51.66

<0.0001

analysed with: Analyse-It + General v1.40

Date

1 February 1999

Hypothesis testing  Research Hypothesis H r --Statement of the researcher’s prediction  Alternate Hypothesis H

a

--Competing explanation of results  Null Hypothesis H

o

-- 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%