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

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

2 DESCRIPTIVE VS. INFERENTIAL STATISTICS Descriptive To organize, summarize & describe the data Inferential To determine reliability of the data

3 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.

4 RELATIONSHIPS SCALES OF Interval Scale MEASURMENT Any of the above plus procedures that include adding. Ratio Scale Any statistical procedure is acceptable.

5 MEASUREMENT SUMMARY Measurement Characteristics Scoring Types Examples Nominal 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. Counting N in sample Labels or # s No relation between # s N of sample Mode Range Football player jerseys 48 not better than 36 Race Gender Must also be mutually exclusive -- one person or item can't fit more than one category.

6 MEASUREMENT SUMMARY Measurement Characteristics Scoring Types Examples Ordinal 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.). Points below which certain % falls. Size of distance between intervals unknown. Order of objects with respect to an attribute. Frequency distribution Median Quartile deviation Spearman rho coefficient of correlation Hardness of metal Personnel evaluations of performance

7 MEASUREMENT SUMMARY Measurement Characteristics Scoring Types Examples Interval Has all of the above characteristics. Added requirement of equal distances or intervals between labels -- represent equal distances in the variables of your study. = intervals w/ arbitrary origin No true zero Adding Mean Standard deviation Variance Pearson product moment coefficient of correlation Temperature difference Footcandle levels in lighting IQ s

8 MEASUREMENT SUMMARY Measurement Characteristics Scoring Types Examples Ratio Has all of above features plus an absolute zero point. Enables you to multiple and divide scale numbers to create ratios between labels. Equal intervals Multiply Divide All types Income ranges. Number of years of school. Age in years. Yardstick or architect s scale.

9 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.

10 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.

11 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.

12 HISTOGRAMS & POLYGONS: GENERAL RULES Place a point at center of score/frequency intersection. Construct either a histogram or polygon.

13 HISTOGRAMS & POLYGONS: Histogram or polygon. GENERAL RULES

14 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

15 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.

16 MEASURES OF CENTRAL Median TENDANCY 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.

17 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.

18 MEASURES OF CENTRAL Mean TENDANCY 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.

19 MEASURES OF CENTRAL TENDANCY Mean (continued) Can calculate the Mean for a distribution of scores or for a frequency distribution. Best indicator of combined performance whereas the median is the best indicator of typical performance.

20 DISTRIBUTION SHAPES - The mean and median are the same. If a single mode, it falls at the same location as the mean and median. SYMMETRICAL

21 DISTRIBUTION SHAPES - When distributions are skewed the values of central tendency differ. Determine skewness by comparing the mean & median without drawing a histogram or polygon. SKEWED

22 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.

23 DISTRIBUTION SHAPES - NEGATIVE SKEW The mean is always smaller than the median & the median is usually smaller than the mode. Skew is to the right.

24 DISTRIBUTION SHAPES - NORMAL CURVE A symmetrical curve with the same number of 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.

25 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.

26 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).

27 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.

28 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.

29 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.

30 Z-scores STANDARD 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.

31 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.

32 Scattergrams CORRELATION 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.

33 Lower left to upper right Positive relationship Low scores on one variable associated with low scores on other High on one high on other. CORRELATION

34 CORRELATION Upper left to lower right Negative relationship. High on one, low on the other variable.

35 CORRELATION Narrow dot band High strength. Straight line shows strong relationship between variables.

36 CORRELATION Scattered dot band Low strength. Relatively weak relationship between variables.

37 CORRELATION Prediction of one variable from another can occur with strong relationships Positive and negative equally important. The higher the correlation between variables in either a positive or negative direction, the more accurate the prediction.

38 CORRELATION COEFFICIENTS Range from to = perfect negative relationship = perfect positive relationship (midpoint) = no relationship at all.

39 CORRELATION COEFFICIENTS Correlation coefficients near unity indicate high degree of relationship. Make accurate prediction about one variable from info about another variable. Desirable to have +/ and above. Again, negative & positive both equally good for prediction.

40 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.

41 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.

42 SPEARMAN RHO Used with rank order data; ordinal scales. Part of the same statistical family as median. Ranges from to (same as Pearsons R).

43 SOURCES OF INFO See your bibliography for the class!

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