What Are Scales of Measurement?

Chapter 3: A Statistics Refresher

What are Scales of Measurement? • Nominal – Categories • Ordinal – Magnitude • Interval – Magnitude and equal interval • Ratio – All others and has a true zero point

Examples • Groups of low, medium, high?

• How much you like something? – 1 not at all 5 very much

Frequency Distribution • Summary information of scores and their occurrence.

Grouped Frequency Distribution • Intervals replace specific values – Identification of 12-15 class intervals recommended

Example of G.F.D. • 20 students surveyed. – Values are the number of alcoholic beverages. – How do you create a frequency distribution using six classes?

Steps • Step 1: Find the highest and lowest values: H = 11 and L = 0. • Step 2: Find the range: R = H – L = 11 – 0 = 11. • Step 3: Select the number of classes desired. In this case: 6. • Step 4: Find the class width by dividing the range by the number of classes. Width = 11/6 = 1.83. Value is rounded up. • Step 5: Select a starting point for the lowest class limit: 0, 2, 4, 6, 8, 10. • Step 6: Upper class limits will be 1, 3, 5, 7, 9, and 11. Most Common Graphs • Histogram • Frequency Polygon • Ogive

Histogram • Intervals = columns; Height = frequency

Frequency Polygon • Intervals points; Plotted at middle.

Ogive • Represents the cumulative frequency.

Measures of Central Tendency • Mean • Median • Mode

Mean • Average score

Mode • Most commonly occurring score • Can you have more than one? • Can you have no mode?

Mode Example • Ten different sports cars were tested for 0-60mph times. • Data set: 4.2, 4.5, 4.5, 4.5, 5, 5.2, 5.5, 5.5, 5.5, 6.2, 6.2

Median (MD) • Rank scores, the median is the middle score . • Number of scores above and below it? – 50th percentile • Even number of scores, median is the average between the two middle scores.

Measures of Variability • Extent of dispersion around central tendency • Variability is useful in interpreting individual differences in the distribution • Range • Variance • Standard Deviation Range • Difference between extreme scores • Highest score is 100, lowest score is 55?

Variance • Variance is equal to the sum of the squared deviations divided by the total number of scores

Variance Problem • Summing the deviations about the mean cancels out the differences (i.e., “0”) • Square the deviations • Squaring the deviation results in positive numbers, though inflated/distorted • What do we do?

Standard Deviation • Square root of the squared deviations about the mean

Standard Deviation Pluses • Expresses individual standings • Shows the dispersion of a distribution • Allows comparisons across independent distributions – As long as they have the same SD

Properties of a Curve • Skewness – Positive and Negative • Kurtosis • Leptokurtic • Mesokurtic: Normal Curve • Properties of a Curve • Platykurtic

What is a Normal Curve? • Bell-shaped curve representing a symmetrical distribution of scores • Mean, mode, median are equal

Figure of a Normal Curve • The normal curve is a frequency polygon

What Does the Area Under the Normal Curve Convey? • Area of the curve describes the proportional distribution of scores • Typically understood in terms of standard deviations from the mean Standard Deviation Cutpoints of a Normal Curve • 50% (median) cuts the distribution in half • 34% is one standard deviation above or below the mean • 68% between -1 and +1 standard deviations • 96% between -2 and +2 standard deviations • 99.7% between -3 and +3 standard deviations

Area Under the Normal Curve • 34% is one standard deviation above or below the mean • 68% between -1 and +1 standard deviations • 47.7% of the curve is between 0 and 1.96 sd units • 95% of the curve is between –1.96 and +1.96 sd units

Skewed Distribution Problems • We can’t normally make comparisons • Mathematically, we can try to normalize (perform a non-linear transformation) the distribution • Convert raw scores to percentile ranks then to z-scores • Not desirable, generally, you want to obtain a normal curve

What are Standard Scores? • Standard scores are linearly transformed scores • Raw scores are mathematically computed that make scores comparable • A standard score distribution maintains the same shape as the original raw score distribution • Best example is the z score

Standard score (z scores) • z score expressed in standard deviation units (0 +/- 1) • What is the z score of a score of 50 on a test with a mean of 30 and a standard deviation of 10?

Standard score (T scores) • T score expressed in standard deviation units as well (50 +/- 10) • What is the T score of a raw score falling 2 standard deviations above the mean would be equal to a T of ???