BHS Methods in Behavioral Sciences I April 18, 2003 Chapter 4 (Ray) – Descriptive Statistics

Scales of Measurement  Nominal (categorical) – all-or-nothing categorization or classification of responses. Example: religions, political parties, occupations  Ordinal – ordered by an underlying continuum, degree of quantitative difference. Example: small, medium, large; child, teen, adult Rank orderings: best, second best, worst

Scales of Measurement (Cont.)  Interval – ordered by a single underlying quantitative dimension with equal intervals between consecutive values. Example: thermometer, rating scales  Ratio – an interval scale with an absolute zero point. Example: height, weight, yearly salary in dollars, heart rate, reaction time to press a button.

Appropriate Statistics  Numbers do not know or care where they came from (how you got them).  It is possible to apply any statistical test to almost any set of numbers, but that doesn’t make it right to do so. Taking the average of football jersey numbers.  It is up to the experimenter to think about the nature of the data when selecting statistics.

Frequency Distributions  Data tells a story.  Techniques for analyzing data help you to figure out what story your data is telling.  Frequency distribution – how frequently does each score appear in your data set. Bar graph Frequency polygon (line graph)

Table 4.2. (p. 87)

Figure 4.1. (p. 88) Bar graph of dream data.

Figure 4.2. (p. 88) Frequency polygon of dream data.

Measures of Central Tendency  What single number best describes the data set? Mean – arithmetical average of a set of scores. Median – the middle score, so that half the numbers are higher and half lower. Mode – the most frequently occurring score.

Figure 4.4. (p. 91) Mean, median, and mode of (a) a normal distribution and (b) a skewed distribution.

Types of Frequency Distributions  Normal – most scores are close to the mean.  Bimodal – the data set has two modes.  Positively skewed – extreme scores in the positive direction  Negatively skewed – extreme scores in the negative direction  In a skewed distribution, the mean is closest to the direction of skew.

Figure 4.3. (p. 89) Four types of frequency distributions: (a) normal, (b) bimodal, (c) positively skewed, and (d) negatively skewed.

Measures of Variability  Variability – how spread out are the scores.  Range – the distance between the highest and lowest scores (largest score minus the smallest scores).  Variance – the average of the squared distances from the mean. Sum of the squares divided by the number of scores.

Figure 4.5. (p. 93) Two different distributions with the same range and mean but different dispersions of scores.

Standard Deviation  Average distance of scores from the mean.  Calculated by taking the square root of the variance. The variance scores were squared so that the average of positive and negative distances from the mean could be combined. Taking the square root reverses this squaring and gives us a number expressed in our original units of measurement (instead of squared units).

Graphing Data  Line graph – used for ordinal, interval, ratio data. Independent variable on the x-axis Dependent variable on the y-axis  Bar graph – used for categorical data.

Figure 4.6. (p. 97) Effects of room temperature on response rates in rats.

Figure 4.7. (p. 97) Effects of different forms of therapy.

Transforming Data  Sometimes it is useful to change the form of the data in some way: Converting F to C temperatures. Converting inches to centimeters.  Transformation lets you compare results across studies.  Transformation must preserve the meaning of the data set and the relationships within it.

Standard Scores  One way to transform data in order to compare two data sets is to express all scores in terms of the distance from the mean. This is called a z-score. z = (score – mean) / standard deviation  z-scores can be transformed so that all scores are positive: This is called a T-score T = 10 x z + 50

Measures of Association  Scatter plot – used to show how two dependent variables vary in relation to each other. One variable on x-axis, the other on y-axis.  Correlation – a statistics that describes the relationship between two variables – how they vary together. Correlations range from -1 to 1.

Figure 4.9. (p. 102) Scatter diagram showing negative relationship between two measures.

Figure (p. 103) Scatter diagrams showing various relationships that differ in degree and direction.