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### Frequency Distributions

• 2. What is a frequency distribution? An organized tabulation of the number of individuals located in each category on a scale of measurement • A method for simplifying and organizing data • Presents an organized picture of the entire set of scores • Are scores generally high or low? • Are the scores clustered together or spread out? • Shows where each individual is located relative to others in a distribution • Where does one score fall relative to all others?
• 3. Frequency Distribution Tables • Consists of at least two columns • Categories on the scale of measurement (X), ordered from lowest to highest • Frequency for each category (how often each category was reported) Original scores: 1, 2, 3, 5, 4, 4, 2, 3, 1, 3, 2, 3, 2, 2 Frequency table: 1 was the lowest reported score. Two people had a score of 1. 2 was the most commonly reported score. Five people had a score of 2.
• 4. Frequency Distributions: Obtaining N • N = the number of observations (or, number of cases) • Remember your notations: • Score = X • Frequency = f • Thus, N = Σf X f 1 2 2 5 3 4 4 2 5 1 Σf = 14 Σf = 2 + 5 + 4 + 2 + 1 Σf = 14
• 5. Frequency Distributions: Obtaining ΣX • Remember your notations: • Score = X • Frequency = f • How do we get ΣX? • Add up all the scores • Multiply each score by the frequency and add up the results X f 1 2 2 5 3 4 4 2 5 1 Σf = 14 Xf 2 x 1 = 2 2 x 5 = 10 3 x 4 = 12 4 x 2 = 8 5 x 1 = 5 ΣX = 37 ΣX = 1 + 1 + 2 + 2 + 2 + 2 + 2 + 3 + 3 + 3 + 3 + 4 + 4 + 5 ΣX = 37
• 6. Obtaining (ΣX)2 and ΣX2 X f 1 2 2 5 3 4 4 2 5 1 Σf = 14 ΣX2 = 12+ 12+ 22+ 22+ 22+ 22+ 22+ 32+ 32+ 32+ 32+ 42+ 42+ 52 ΣX2 = 1 + 1 + 4 + 4 + 4 + 4 + 4 + 9 + 9 + 9 + 9 + 16 + 16 + 25 ΣX2 = 115 • Remember the order of operations: 1. Parentheses 2. Exponents 3. Multiplication/Division 4. Adding/Subtracting (ΣX)2 = (37)2 (ΣX)2 =1369
• 7. Proportions (p) and Percentages (%) • p measures the fraction of the total group associated with each score • p = f/N X f 1 2 2 5 3 4 4 2 5 1 Σf = 14 p = f/N 2/14 = 0.143 5/14 = 0.357 4/14 = 0.286 2/14 = 0.143 1/14 = 0.071 % = p(100) 14.3% 35.7% 28.6% 14.3% 7.1% Σp = 1.00 Σ% = 100% *Math check: sum proportions = 1.00; sum percentages = 100%
• 8. Types of Frequency Distribution Tables Regular Frequency Distribution • Lists all of the individual categories (X values) Grouped Frequency Distribution • When listing all of the individual categories is not possible/helpful/reasonabl e • e.g., Test scores 0-100 X f 1 2 2 5 3 4 4 2 5 1 Σf = 14
• 9. Grouped Frequency Tables • Range of scores on a test: 39 – 85 • Organized by class intervals • More efficient • Guidelines 1. No more than 10 class intervals 2. Width of each interval should be a simple number (i.e., 2, 5, 10, 20) 3. Bottom score of each interval should be a multiple of the width (in this case, a multiple of “10”) 4. All intervals should be equal X f 80-89 4 70-79 6 60-69 12 50-59 8 40-49 4 30-39 2
• 10. Grouped Frequency Tables • What information is lost? • The individual scores – and thus, our ability to solve for Σ(Xf) X f 80-89 4 70-79 6 60-69 12 50-59 8 40-49 4 30-39 2
• 11. Constructing a Grouped Frequency Table For a set of N = 25 scores: 82, 75, 88, 93, 53, 84, 87, 58, 72, 94, 69, 84, 61, 91, 64, 87, 84, 70, 76, 89, 75, 80, 73, 78, 60
• 12. Constructing a Grouped Frequency Table For a set of N = 25 scores: 82, 75, 88, 93, 53, 84, 87, 58, 72, 94, 69, 84, 61, 91, 64, 87, 84, 70, 76, 89, 75, 80, 73, 78, 60 1. Determine the range of scores • Range = 42 Smallest score is X = 53 Largest score is X = 94 94 - 53 = 42
• 13. Constructing a Grouped Frequency Table For a set of N = 25 scores: 82, 75, 88, 93, 53, 84, 87, 58, 72, 94, 69, 84, 61, 91, 64, 87, 84, 70, 76, 89, 75, 80, 73, 78, 60 1. Determine the range of scores • Range = 42 2. Group scores into intervals • At least 9 intervals needed Smallest score is X = 53 Largest score is X = 94 94 - 53 = 42 Width # of Intervals Needed to Cover a Range of 42 Points 2 21 (too many) 5 9 (OK) 10 5 (too few)
• 14. Constructing a Grouped Frequency Table For a set of N = 25 scores: 82, 75, 88, 93, 53, 84, 87, 58, 72, 94, 69, 84, 61, 91, 64, 87, 84, 70, 76, 89, 75, 80, 73, 78, 60 1. Determine the range of scores • Range = 42 2. Group scores into intervals • At least 9 intervals needed 3. Identify the intervals • Bottom interval should be 50-54 • Next is 55-59 • Then 60-64 • Etc. Smallest score is X = 53 Largest score is X = 94 94 - 53 = 42 Lowest score is X = 53 Lowest interval should contain this value Interval should have a multiple of 5 as its bottom score Interval should contain 5 values (because it has an interval of 5)
• 15. Constructing a Grouped Frequency Table For a set of N = 25 scores: 1. Determine the range of scores • Range = 42 2. Group scores into intervals • At least 9 intervals needed 3. Identify the intervals • Bottom interval should be 50-54 • Next is 55-59 • Then 60-64 • Etc. X 90-94 85-89 80-84 75-79 70-74 65-69 60-64 55-59 50-54 f 3 4 5 4 3 1 3 1 1 p = f/N 0.12 0.16 0.20 0.16 0.12 0.04 0.12 0.04 0.04 Σp 1.00 Σ% 100% % 12% 16% 20% 16% 12% 4% 12% 4% 4%
• 17. Real Limits and Frequency Distributions X 90-94 85-89 80-84 75-79 70-74 65-69 60-64 55-59 50-54 f 3 4 5 4 3 1 3 1 1 • For X = 60 – 64, there are 3 scores • Does not mean 3 scores are identical • Means 3 scores fall within the interval • For each interval, there are limits • Lower real limit • Upper real limit 3 different scores within this interval 50 55 60 65 70 75 80 85 90 95 59.5 – 64.5 Apparent Limits Real Limits
• 18. Frequency Distribution Graphs • X-axis (abscissa) • Horizontal line • Values increase from left to right • Y-axis (ordinate) • Vertical line • Values increase from bottom to top • In a frequency distribution graph: • Score categories (X values) are listed on the X axis • Frequencies are listed on the Y axis
• 19. Graphs for Interval or Ratio Data Histogram • Adjacent bars touch • Height of bar indicates frequency • Width corresponds to the score (or limits of the range of scores) Polygon • Vertical position of dot indicates a score’s frequency • Continuous line is draw between series of dots • X = 0 typically one category above/below highest/lowest score
• 20. A Modified Histogram • Each individual is represented by a block placed directly above the individual’s score. How many people had scores of X = 2?
• 21. Histogram and Polygon on One Graph
• 22. What If Your Data Is Not Interval/Ratio? • Bar Graph • For nominal or ordinal data • Like a histogram in structure, but the bars do not touch • Emphasizes distinct categories
• 24. Don’t Misrepresent Your Data! Leave NYC immediately! Oh, wait… A Warning…
• 25. Graphs for Population Distributions Relative Frequencies • We may not know exactly how many fish are in Lake Erie, but we do know that there are double the number of Bluegill than there are Bass Smooth Curves • When the population consists of numerical scores from an interval or ratio scale • Indicates that we are showing the relative changes that occur from one score to the next
• 26. Describing a Frequency Distribution • Three characteristics completely describe any distribution • Central tendency (Chapter 3) • Where is the center of the distribution? • Variability (Chapter 4) • Are the scores spread out, or clustered together? • Shape • Are the scores normally distributed among the population?
• 27. The Shape of a Frequency Distribution
• 28. Skewed Distributions Positively Skewed • Scores tend to pile up on the left side • Tail “points” to the right” • The “skew” is on the “positive” side of the curve Negatively Skewed • Scores tend to pile up on the right side • Tail “points” to the left • The “skew” is on the “negative” side of the curve
• 30. Name That Distribution! Symmetric (no skew) Positive Skew Negative Skew
• 31. Percentiles & Percentile Ranks • Percentile rank • The percentage of individuals in the distribution with scores at or below the particular value • Percentile • When a score is identified by its percentile rank For example: o Your exam score is X = 43 o 60% of the class had scores of 43 or lower o Your score has a percentile rank of 60% and is called the 60th percentile.
• 32. Percentiles & Percentile Ranks X f Xf p % cf c% 1 2 2 0.143 14.3% 2 (2/14)100% = 14.3% 2 5 10 0.357 35.7% (2+5) = 7 (7/14)100% = 50% 3 4 12 0.286 28.6% (7+4) = 11 (11/14)100% = 78.6% 4 2 8 0.143 14.3% (11+2) = 13 (13/14)100% = 92.9% 5 1 5 0.071 7.1% (13+1) = 14 (14/14)100% = 100% N = 14 (x lowx high) (cf/N)(100%) • To find percentiles and percentile ranks, we must first calculate cumulative frequency (cf) and cumulative percentage (c%) for each score (or interval)
• 33. Let’s Practice 1. Find the 70th percentile (remember real limits!) 2. Find the percentile rank for X = 9.4 X f cf c% 0 – 4 2 2 10% 5 – 9 4 6 20% 10 – 14 8 14 70% 15 – 19 5 19 95% 20 - 24 1 20 100%
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