DESCRIPTIVE STATISTICS

The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses.

DESCRIPTIVE VS. INFERENTIAL STATISTICS

• Descriptive– To organize, summarize &

describe the data

• Inferential– To determine reliability of the data

RELATIONSHIPS – SCALES OF MEASURMENT

• Nominal Scale– Only use those statistical procedures

that rely on counting -- the number (N) in the sample.

• Ordinal Scale– Same as nominal scale– Can use statistics that indicate points

below which certain percentages of the cases fall.

RELATIONSHIPS – SCALES OF MEASURMENT

• Interval Scale– Any of the above plus procedures

that include adding.

• Ratio Scale– Any statistical procedure is

acceptable.

MEASUREMENT SUMMARY

Football player jerseys – 48 not better than 36

Race

Gender

N of sample

Mode

Range

Counting“N” in sample

Labels or #’s

No relation between #’s

Lowest level -- used to classify variables into two or more categories.Cases placed in the same category must be equivalent. The categories must be exhaustive -- all persons or items must fit into one of the categories.Must also be mutually exclusive -- one person or item can't fit more than one category.

Nominal

ExamplesTypesScoringCharacteristicsMeasurement

MEASUREMENT SUMMARY

ExamplesTypesScoringCharacteristicsMeasurement

Hardness of metal

Personnel evaluations of performance

Frequency distribution

Median

Quartile deviation

Spearman rho coefficient of correlation

Points below which certain % falls.

Size of distance between intervals unknown.

Order of objects with respect to an attribute.

Numbers only used to indicate the rank order of cases of a variable. Cannot measure or evaluate the difference in value between each case.No mathematical or statistical operations (you can't add label 1 to label 2, etc.).

Ordinal

MEASUREMENT SUMMARY

ExamplesTypesScoringCharacteristicsMeasurement

Temperature difference

Footcandlelevels in lighting

IQ’s

Mean

Standard deviation

Variance

Pearson product moment coefficient of correlation

= intervals w/ arbitrary origin

No true zero

Adding

Has all of the above characteristics.

Added requirement of equal distances or intervals between labels --represent equal distances in the variables of your study.

Interval

MEASUREMENT SUMMARY

ExamplesTypesScoringCharacteristicsMeasurement

Income ranges.

Number of years of school.

Age in years.

Yardstick or architect’s scale.

All typesEqual intervals

Multiply

Divide

Has all of above features plus an absolute zero point.

Enables you to multiple and divide scale numbers to create ratios between labels.

Ratio

FREQUENCY DISTRIBUTIONS

• The arrangement of the scores from lowest to highest.

• Implies a general shape to the data because of the shape of the distribution.

FREQUENCY DISTRIBUTIONS

• The easiest way for you to do summary statistics is with a dedicated statistical package.

• With small data sets, you can do most data manipulation for summary statistics with a spreadsheet.

HISTOGRAMS & POLYGONS: GENERAL RULES

• On horizontal axis, lay out lowest scores to highest -- left to right.

• Lay out frequencies on vertical axis --from 0 up to highest frequency.

HISTOGRAMS & POLYGONS: GENERAL RULES

• Place a point at center of score/frequency intersection.

• Construct either a histogram or polygon.

HISTOGRAMS & POLYGONS: GENERAL RULES

• Histogram or polygon.

MEASURES OF CENTRAL TENDANCY

• Used to summarize data through a single number that can represent the whole set of scores.

• Types: mode, median, mode, mean

MEASURES OF CENTRAL TENDANCY

• Mode– The value or number that occurs

most frequently in the distribution. Two modes are bi-modal; three or more are tri-model or multi-modal.

– Very stable and there can be more than one mode.

– Only appropriate measure for nominal scales.

MEASURES OF CENTRAL TENDANCY

• Median– The point in the distribution below which

50% of the scores lie. – Scores must be placed in rank order

from lowest to highest first. – The median can fall between the upper

limit and lower limit of a score. – Can fall on the border line between

scores.

MEASURES OF CENTRAL TENDANCY

• Median (continued)– The median is an ordinal statistic

because it is based on rank. – Can be used on interval and ratio

data but the interval characteristic of the data is not used.

– Only time the median is really useful is when there are extreme scores in the distribution.

MEASURES OF CENTRAL TENDANCY

• Mean

– The arithmetic average --sum of all the scores divided by the N.

– Most stable measure of central tendency and is more precise than the median or mode.

– Can be used with interval and ratio scales.

MEASURES OF CENTRAL TENDANCY

• Mean (continued)– Can calculate the Mean for a

distribution of scores or for a frequency distribution.

– Best indicator of combinedperformance whereas the median is the best indicator of typicalperformance.

DISTRIBUTION SHAPES -SYMMETRICAL

• The mean and median are the same.

• If a single mode, it falls at the same location as the mean and median.

DISTRIBUTION SHAPES -SKEWED

• When distributions are skewed the values of central tendency differ.

• Determine skewness by comparing the mean & median without drawing a histogram or polygon.

DISTRIBUTION SHAPES -POSITIVE SKEW

• The mean is always greater than the median & the median is usually greater than the mode.

• Skew is to the left.

DISTRIBUTION SHAPES -NEGATIVE SKEW

• The mean is always smallerthan the median & the median is usually smaller than the mode.

• Skew is to the right.

DISTRIBUTION SHAPES -NORMAL CURVE

• A symmetrical curve with the same numberof scores above & below the mean.

• Same as symmetrical. • Most scores are

concentrated around the mean.

• Approximately 68% of the cases are within +/- 1 SD unit from the mean.

VARIABILITY MEASURES

• Range– Difference between the highest

and lowest scores. – Determine by subtraction. – Is an unreliable index of variability

because it is derived from only two scores.

VARIABILITY MEASURES• Quartile deviation

– Half the difference between the upper and lower quartiles in a distribution.

– The 75th percentile & the 25th percentile.

– Provides a measure of one-half of the range of scores within which lie the middle 50% of the scores.

– It is an ordinal scale statistic and is used with the median (which means that it is not often used unless there are extreme scores).

VARIABILITY MEASURES• Variance

– Based on the mean.– Considers the size and location of

individual scores. – Variance & standard deviation are based

on the deviation score which is the difference between a raw score & the mean.

– The sum of the deviation scores of a distribution are always zero because the scores above the mean are always positive while the scores below the mean are always negative.

VARIABILITY MEASURES

• Standard Deviation– SD is the square root of variance – Is used to summarize data in the

same units as the original data. – Most commonly used statistic for

variability. – It is the square root of the mean of

the squared deviation scores.

STANDARD SCORES

• z-scores– The distance of a score from the mean

in standard deviation units. – Scores with the same numerical value

as the mean will have a z-score of zero. – Used to compare one set of scores to

another -- example two exams and S's performance on the exams.

– Use of z-scores requires use of negative values and fractions. Overcome by using Z-scores.

STANDARD SCORES

• Z-scores– Obtained by multiplying the z-score by

10 and adding 50 to the result. – Used to compare scores in different

distributions. – Allows descriptions in whole numbers.– A type of standard score. – Does not alter the shape of the original

distribution.

CORRELATION

• Used to describe the relationship between pairs of scores.

• Shows the extent to which a change in one variable is associated with change in another variable.

CORRELATION• Scattergrams

– Used to show correlation. – One variable on each axis (horizontal

and vertical). – Plot scattergrams to see both direction

& strength of a relationship. – Direction shows positive or negative

relationship.– Scores for independent variable on

horizontal axis & dependent variable on vertical axis.

CORRELATION

• Lower left to upper right– Positive

relationship– Low scores on

one variable associated with low scores on other

– High on one highon other.

CORRELATION

• Upper left to lower right– Negative

relationship.– High on one, low

on the other variable.

CORRELATION

• Narrow dot band– High strength.– Straight line

shows strong relationship between variables.

CORRELATION

• Scattered dot band– Low strength.– Relatively weak

relationship between variables.

CORRELATION• Prediction of one variable from

another can occur with strong relationships

• Positive and negative equallyimportant.

• The higher the correlation between variables in either a positive or negative direction, the more accurate the prediction.

CORRELATION COEFFICIENTS

• Range from -1.00 to +1.00. • -1.00 = perfect negative

relationship.• +1.00 = perfect positive

relationship. • 0.00 (midpoint) = no relationship

at all.

CORRELATION COEFFICIENTS

• Correlation coefficients near unityindicate high degree of relationship.

• Make accurate prediction about one variable from info about another variable.

• Desirable to have +/- 0.90 and above.• Again, negative & positive both

equally good for prediction.

PEARSONS R(PRODUCT MOMENT

CORRELATION)• Used with either interval or

ratio scales. • Defined as the mean of z-score

products of two variables.• Most common method for

correlation.• Same statistical family as

mean.

PEARSONS R(PRODUCT MOMENT

CORRELATION)

• Assumes a linear relationship between the two variables. (Straight line fit between scores of the two variables).

• If curvilinear, must use other methods.

SPEARMAN RHO

• Used with rank order data; ordinal scales.

• Part of the same statistical family as median.

• Ranges from -1.00 to +1.00 (same as Pearsons R).

SOURCES OF INFO

• See your bibliography for the class!