Chapter 2 Describing and Presenting a Distribution of Scores

Chapter Objectives After completing this chapter, you should be able to 1. Define all statistical terms that are presented. 2. Describe the four scales of measurement and provide examples of each. 3. Describe a normal distribution and four curves for distributions that are not normal. 4. Define the terms measures of central tendency and measures of variability. 5. Define the three measures of central tendency, identify the symbols used to represent them, describe their characteristics, calculate them with ungrouped data, and state how they can be used to interpret data. 2-2

Chapter Objectives 6. Define the three measures of variability, identify the symbols used to represent them, describe their characteristics, calculate them with ungrouped data, and state how they can be used to interpret data. 7. Define percentile and percentile rank, identify the symbols used to represent them, calculate them with ungrouped date, and state how they can be used to interpret data. 8. Define standard scores, calculate z-scores, and interpret their meanings. 2-3

Statistical Terms descriptive statistics inferential statistics discrete data continuous data ungrouped data data variable population sample random sample parameter statistic (as contrasted with statistics; page 1 of textbook) 2-4

Nominal Scale - lowest and most elementary; also called categorical; naming level only; no comparisons of categories Ordinal Scale - order or rank; no indication of how much better one score is than other Interval Scale - order or rank; same distance exists between each division; no true zero Ratio Scale - possesses all characteristics of interval scale and has true zero point Scales of Measurement (See Table 2.1) 2-5

Normal Distribution (See Figure 2.1) Most statistical methods are based on assumption that a distribution of scores is normal and that the distribution can be graphically represented by the normal curve (bell-shaped). Normal distribution is theoretical and is based on the assumption that the distribution contains an infinite number of scores. 2-6

Characteristics of Normal Curve Bell-shaped curve Symmetrical distribution about vertical axis of curve Greatest number of scores found in middle of curve All measures of central tendency at vertical axis 2-7

mean median mode 2-8

Different Curves (see Figure 2.2) leptokurtic – very similar in ability; homogeneous group platykurtic – wide range of ability; heterogeneous group bimodal - two high points skewed - scores clustered at on end; positive or negative 2-9

Analysis of Ungrouped Data Better understanding of data Interpret data 2-10

Score Rank 1. List scores in descending order. 2. Number the scores; highest score is number 1 and last score is the number of the total number of scores. 3. Average rank of identical scores and assign them the same rank (may determine the midpoint and assign that rank). 2-11

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Measures of Central Tendency descriptive statistics describe the middle characteristics of the data (distribution of scores); represent scores in a distribution around which other scores seem to center most widely used statistics mean, median, and mode 2-13

Mean Characteristics Most sensitive of all measures of central tendency Most appropriate measure of central tendency to use for ratio data (may be used on interval data) Considers all information about the data and is used to perform other statistical calculations Influenced by extreme scores, especially if the distribution is small The arithmetic average of a distribution of scores; most generally used measure of central tendency. 2-14

Symbols Used to Calculate Mean X = the mean (called X-bar)  = (Greek letter sigma) = “the sum of” X = individual score N = the total number of scores in distribution Mean Formula X =  X N Table 2.3: X = 2644 =

Median Characteristics Not affected by extreme scores. A measure of position. Not used for additional statistical calculations. Represented by Mdn or P 50. Score that represents the exact middle of the distribution; the fiftieth percentile; the score that 50% of the scores are above and 50% of the scores are below. 2-16

Steps In Calculation of Median 1. Arrange the scores in ascending order. 2. Multiple N by If the number of scores is odd, P 50 is the middle score of the distribution. 4. If the number of scores is even, P 50 is the arithmetic average of the two middle scores of the distribution. Table 2.3:.50(30) = 15 Fifteenth and sixteenth scores are 88 P 50 =

Mode Score that occurs most frequently; may have more than one mode. Characteristics Least used measure of central tendency. Not used for additional statistics. Not affected by extreme scores. Table 2.3: Mode =

Which Measure of Central Tendency is Best for Interpretation of Test Results? (See Figure 2.3) Mean, median, and mode are the same for a normal distribution, but often will not have a normal curve. The farther away from the mean and median the mode is, the less normal the distribution. The mean and median are both useful measures. In most testing, the mean is the most reliable and useful measure of central tendency; it is also used in many other statistical procedures. 2-19

Measures of Variability To provide a more meaningful interpretation of data, you need to know how the scores spread. Variability - the spread, or scatter, of scores; terms dispersion and deviation often used With the measures of variability, you can determine the amount that the scores spread, or deviate, from the measures of central tendency. Descriptive statistics; reported with measures of central tendency 2-20

Range Determined by subtracting the lowest score from the highest score; represents on the extreme scores. Characteristics 1. Dependent on the two extreme scores. 2. Least useful measure of variability. Formula: R = H x - L x Table 2.3: R = =

Quartile Deviation Sometimes called semiquartile range; is the spread of middle 50% of the scores around the median. Extreme scores will not affect the quartile deviation. Characteristics 1. Uses the 75th and 25th percentiles; difference between these two percentiles is referred to as the interquartile range. 2. Indicates the amount that needs to be added to, and subtracted from, the median to include the middle 50% of the scores. 3. Usually not used in additional statistical calculations. 2-22

Quartile Deviation Symbols Q = quartile deviation Q 1 = 25th percentile or first quartile (P 25 ) = score in which 25% of scores are below and 75% of scores are above Q 3 = 75th percentile or third quartile (P 75 ) = score in which 75% of scores are below and 25% of scores are above 2-23

Steps for Calculation of Q 3 1. Arrange scores in ascending order. 2. Multiply N by.75 to find 75% of the distribution. 3. Count up from the bottom score to the number determined in step 2. Approximation and interpolation may be required. Steps for Calculation of Q 1 1. Multiply N by.25 to find 25% of the distribution. 2. Count up from the bottom score to the number determined in step 1. To Calculate Q Substitute values in formula: Q = Q 3 - Q

Q 1 = 25% Q 2 = 50% Q 3 = 75% Q 4 = 100% Q 2 - Q 1 = range of scores below median Q 3 - Q 2 = range of scores above median Quartiles 2-25

Table 2.3: 1..75(30) = 22.5; twenty-second score = 90; twenty-third score = 90; midway between two scores would be same score 75% = (30) = 7.5; seventh score = 85; eight score = 86; midway between two scores = Q = = 4.5 = Table 2.3: = = Theoretically, middle 50% of scores fall between the scores of and

Standard Deviation Most useful and sophisticated measure of variability. Describes the scatter of scores around the mean. Is a more stable measure of variability than the range or quartile deviation because it depends on the weight of each score in the distribution. Lowercase Greek letter sigma (σ) is used to indicate the standard deviation of a population; letter s is used to indicate the standard deviation of a sample. Since you generally will be working with small samples, the formula for determining the standard deviation will include (N - 1) rather than N. 2-27

Characteristics of Standard Deviation 1. Is the square root of the variance, which is the average of the squared deviations from the mean. Population variance is represented as σ 2 and the sample variance is represented as s Is applicable to interval and ratio data, includes all scores, and is the most reliable measure of variability. 3. Is used with the mean. In a normal distribution, one standard deviation added to the mean and one standard deviation subtracted from the mean includes the middle 68.26% of the scores. 2-28

Characteristics of Standard Deviation 4. With most data, a relatively small standard deviation indicates that the group being tested has little variability (performed homogeneously). A relatively large standard deviation indicates the group has much variability (performed heterogeneously). 5. Is used to perform other statistical calculations. Symbols used to determine the standard deviation: s = standard deviation X = individual score X = mean N = number of scores Σ = sum of d = deviation score (X - X) 2-29

Calculation of Standard Deviation with ΣX 2 1. Arrange scores into a series. 2. Find ΣX Square each of the scores and add to determine the ΣX Insert the values into the formula NΣX 2 - (ΣX) 2 s = N(N- 1) Table 2.3: ΣX = 2644 N = 30 ΣX 2 = 233,398 s =

Calculation of Standard Deviation with Σd 2 1. Arrange the scores into a series. 2. Calculate X. 3. Determine d and d 2 for each score; calculate Σd Insert the values into the formula Σd 2 s = N - 1 Table 2.4: X = 88.1 s = 3.6 Σd 2 = N =

Interpretation of Standard Deviation in Tables 2.3 and 2.4 S = 3.6 X = = = 84.5 In a normal distribution, 68.26% of the scores would fall between 84.5 and

Relationship of Standard Deviation and Normal Curve (See Figure 2.4) Based on the probability of a normal distribution, there is an exact relationship between the standard deviation and the proportion of area and scores under the curve % of the scores will fall between +1.0 and -1.0 standard deviations % of the scores will fall between and standard deviations % of the scores will fall between +3.0 and standard deviations. 4. Generally, scores will not exceed +3.0 and -3.0 standard deviations from the mean. 2-33

Figure 2.4 Characteristics of normal curve. 2-34

60-sec Sit-up Test to Two Fitness Classes Class 1Class 2 X = 32X = 28 s = 2s = 4 Figure 2.5 compares the spread of the two distributions. Individual A in Class 1 completed 34 sit-ups and individual B completed 34 sit-ups in Class 2. Both individuals have the same score, but do not have the same relationship to their respective means and standard deviations. Figure 2.6 compares the individual performances. 2-35

Calculation of Percentile Rank through Use of Mean and Standard Deviation. 1. Calculate the deviation of the score from the mean. d = (X - X) 2. Calculate the number of standard deviation units the score is from the mean (z-scores). No. of standard deviation units from the mean = d s 3. Use table 2.5 to determine where the percentile rank of the score is on the curve. If negative value found in step 1, the percentile rank will always be less than

Which Measure of Variability is Best for Interpretation of Test Results? 1. Range is the least desirable. 2. The quartile deviation is more meaningful than the range, but it considers only the middle 50% of the scores. 3. The standard deviation considers every score, is the most reliable, and is the most commonly used measure of variability. 2-37

Percentiles and Percentile Ranks Percentile - a point in a distribution of scores below which a given percentage of scores fall. Examples - 60th percentile and 40 percentile Percentile rank - percentage of the total scores that fall below a given score in a distribution; determined by beginning with the raw scores and calculating the percentile ranks for the scores. 2-38

Weakness of Percentiles 1.The relative distance between percentile scores are the same, but the relative distances between the observed scores are not. 2. Since percentile scores are based on the number of scores in a distribution rather than the size of the score obtained, it is sometimes more difficult to increase a percentile score at the ends of the scale than in in the middle. 3. Average performers (in middle of distribution) need only a small change in their raw scores to produce a large change in their percentile scores. 4. Below average and above average performers (at ends of distribution) need a large change in their raw scores to produce even a small change in their percentile scores. 2-39

Frequency Distribution Serve to group data Scores listed in ascending or descending order Number of times each occurs indicated Percent of times each score occurs indicated Cumulative percent (percent of scores below a given score) indicated See table

Graphs 1. Enable individuals to interpret data without reading raw data or tables. 2. Different types of graphs are used. Examples - histogram (column), frequency polygon (line), pie chart, area, scatter, and pyramid 3. Standard guidelines should be used when constructing graphs. See figures 2.7, 2.8, 2.9, and

Standard Scores Provide method for comparing unlike scores; can obtain an average score, or total score for unlike scores. z-score - represents the number of standard deviations a raw score deviated from the mean FORMULA z = X - X s 2-42

Table 2.7 – Tennis Serve Scores Scores of 88 and 54; X = 72.2; s = 10.8 z = X - X s z = = 15.9 z = = z = 1.47 z = INTERPRETION? z-Scores 2-43

z-Scores The z-scale has a mean of 0 and a standard deviation of 1. Normally extends from –3 to +3 standard deviations. All standard scored are based on the z-score. Since z-scores are expressed in small, involve decimals, and may be positive or negative, many testers do not use them. Table 2.5 shows relationship of standard deviation units and percentile rank. 2-44

T-Scores T-scale Has a mean of 50. Has a standard deviation of 10. May extend from 0 to 100. Unlikely that any t-score will be beyond 20 or 80 (this range includes plus and minus 3 standard deviations). Formula T-score = (X - X) = z s Figure 2.11 shows the relationship of z-scores, T-scores, and the normal curve. 2-45

Figure 2.9 z-scores and T-scores plotted on normal curve. 2-46

T-Scores Table Tennis Serve Scores Scores of 88 and 54; X = 72.2; s = 10.8 T 88 = (1.47) T 54 = (-1.68) = = 50 + (-16.8) = 64.7 = 65 = 33.2 = 33 (T-scores are reported as whole numbers) 2-47

T-Scores T-scores may be used in same way as z-scores, but usually preferred because: Only positive whole numbers are reported. Range from 0 to 100. Sometime confusing because 60 or above is good score. 2-48

T-Scores May convert raw scores in a distribution to T-scores See Table Number a column of T-scores from 20 to Place the mean of the distribution of the scores opposite the T-score of Divide the standard deviation of the distribution by 10. The standard deviation for the T-scale is 10, so each T-score from 0 to 100 is one-tenth of the standard deviation. 2-49

T-Scores 4. Add the value found in step 3 to the mean and each subsequent number until you reach the T-score of Subtract the value found in step 3 from the mean and each decreasing number until you reach the number Round off the scores to the nearest whole number. *For some scores, lower scores are better (timed events). 2-50

Percentiles Are standard scores and may be used to compare scores of different measurements. Change at different rates (remember comparison of low and and high percentile scores with middle percentiles), so they should not be used to determine one score for several different tests. May prefer to use T-scale when converting raw scores to standard scores. 2-51