Lesson 4 Measures of Variability and Position PDF

Lesson 4: Measures of Variability and Position Objectives: 1. Delineate the purpose of using measures of variability. 2. Define and calculate the range as a simple measure of variability and explain its limitations. 3. Differentiate between standard deviation and variance. 4. Com...

Lesson 4: Measures of Variability and Position Objectives: 1. Delineate the purpose of using measures of variability. 2. Define and calculate the range as a simple measure of variability and explain its limitations. 3. Differentiate between standard deviation and variance. 4. Compute for quartile, decile, and percentile as measures of position. A. MEASURES OF VARIABILITY The term variability has much the same meaning in statistics as it has in everyday language; to say that things are variable means that they are not all the same. In statistics, our goal is to measure the amount of variability for a particular set of scores, a distribution. In simple terms, if the scores in a distribution are all the same, then there is no variability. If there are small differences between scores, then the variability is small (homogenous), and if there are large differences between scores, then the variability is large (heterogenous). Variability provides a quantitative measure of the differences between scores in a distribution and describes the degree to which the scores are spread out or clustered together. Refer to the table below to familiarize symbols used to describe values. STANDARD TOTAL NUMBER VARIANCE MEAN DEVIATION OF SCORES POPULATION µ N SAMPLE s s2 n The figure below shows two distributions of familiar values for the population of adult males: part (a) shows the distribution of men’s heights (in inches), and part (b) shows the distribution of men’s weights (in pounds). Notice that the two distributions differ in terms of central tendency. The mean height is 70 inches (5 feet, 10 inches) and the mean weight is 170 pounds. In addition, notice that the distributions differ in terms of variability. For example, most heights are clustered close together, within 5 or 6 inches of the mean. On the other hand, weights are spread over a much wider range. In the weight distribution it is not unusual to find individuals who are located more than 30 pounds away from the mean, and it would not be surprising to find two individuals whose weights differ by more than 30 or 40 pounds.  The purpose for measuring variability is to obtain an objective measure of how the scores are spread out in a distribution. In general, a good measure of variability serves two purposes: 1. Variability describes the distribution. Specifically, it tells whether the scores are clustered close together or are spread out over a large distance. Usually, variability is defined in terms of distance. It tells how much distance to expect between one score and another, or how much distance to expect between an individual score and the mean. For example, we know that the heights for most adult males are clustered close together, within 5 or 6 inches of the average. Although more extreme heights exist, they are relatively rare. 2. Variability measures how well an individual score (or group of scores) represents the entire distribution. This aspect of variability is very important for inferential statistics, in which relatively small samples are used to answer questions about populations. For example, suppose that you selected a sample of one person to represent the entire population. Because most adult males have heights that are within a few inches of the population average (the distances are small), there is a very good chance that you would select someone whose height is within 6 inches of the population mean. On the other hand, the scores are much more spread out (greater distances) in the distribution of weights. In this case, you probably would not obtain someone whose weight was within 6 pounds of the population mean. Thus, variability provides information about how much error to expect if you are using a sample to represent a population. Range Range is the distance covered by the scores in a distribution, from the smallest score to the largest score. Formula: Range = Highest score – Lowest score EXAMPLE: Find the range of the following scores: 23,28,17,18,20 Range = 28 – 17 Range = 11 Characteristics of the Range The range is probably the most obvious way to describe how spread out the scores are—simply find the distance between the maximum and the minimum scores. The problem with using the range as a measure of variability is that it is completely determined by the two extreme values and ignores the other scores in the distribution. Thus, a distribution with one unusually large (or small) score will have a large range even if the other scores are all clustered close together. Because the range does not consider all the scores in the distribution, it often does not give an accurate description of the variability for the entire distribution. For this reason, the range is a crude and unreliable measure of variability. Standard Deviation The standard deviation is the most used and the most important measure of variability. Standard deviation uses the mean of the distribution as a reference point and measures variability by considering the distance between each score and the mean. In simple terms, the standard deviation provides a measure of the standard, or average, distance from the mean, and describes whether the scores are clustered closely around the mean or are widely scattered. A. RAW DATA 1. Raw Score Method FORMULA: 2 ∑ 𝑋 2 − (∑ 𝑋) √ 𝑛 𝑠= 𝑛−1 Where: 𝑛 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑐𝑜𝑟𝑒𝑠 𝑠 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 ∑ 𝑋 2 = 𝑠𝑢𝑚𝑚𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑞𝑢𝑎𝑟𝑒𝑑 𝑠𝑐𝑜𝑟𝑒𝑠 ∑ 𝑋 = 𝑠𝑢𝑚𝑚𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑐𝑜𝑟𝑒𝑠  Example: Scores on Quiz Squared score (X) X2 14 196 15 225 n=5 10 100 7 49 8 64 ∑X = 54 ∑X2 = 634 Step 1: Compute for ∑X by adding all scores Step 2: Square each score (X2) Step 3: Compute for ∑X2 by adding all squared scores Step 4: Use the formula to substitute the needed values 2 (∑ 𝑋)2 √∑ 𝑋 − 𝑛 𝑠= 𝑛−1 542 √634 − 5 𝑠= 5−1 𝑠 = √12.7 𝑠 = 3.56 * The average distance of the scores from the mean is 3.56. B. ORGANIZED DATA Midpoint Method FORMULA: 𝑛(∑ 𝑓𝑋 2 ) − (∑ 𝑓𝑋)2 𝑠= √ 𝑛 (𝑛 − 1) Where: 𝑛 = 𝑡𝑜𝑡𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑖𝑒𝑠 𝑠 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 ∑ 𝑓𝑋 2 = 𝑠𝑢𝑚𝑚𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 (𝑋)(𝑓𝑋) ∑ 𝑓𝑋 = 𝑠𝑢𝑚𝑚𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑓𝑋  Example: X (Midpoint of Class Intervals f each class fX fX 2 interval) (X)(fX) 15-17 4 16 64 1024 12-14 6 13 78 1014 9-11 8 10 80 800 6-8 12 7 84 588 3-5 10 4 40 160 ∑f = 40 (N) ∑fX = 346 ∑ fX = 3586 2 Step 1: Compute for n by adding all frequencies Step 2: Compute for the midpoint of each class interval using the formula 𝐿𝐿 + 𝑈𝐿 2 Step 3: Multiply each class interval’s frequency with corresponding midpoint (fX) Step 4: Add all fX to get its summation (∑fX) Step 5: Multiply each class interval’s midpoint with fX to get fX 2 Step 6: Add all the values for fX 2 to get its summation (∑ fX 2 ) Step 7: Use the formula to substitute the needed values 𝑛(∑ 𝑓𝑋 2 ) − (∑ 𝑓𝑋)2 𝑠= √ 𝑛 (𝑛 − 1) 40 (3586) − (346)2 𝑠= √ 40 (40 − 1) 𝑠 = 3.90 * The standard distance of the scores from the mean is 3.90 Characteristics of the Standard Deviation The standard deviation has many important characteristics. First, the standard deviation gives us a measure of dispersion relative to the mean. This differs from the range, which gives us an absolute measure of the spread between the two most extreme scores. Second, the standard deviation is sensitive to each score in the distribution. If a score is moved closer to the mean, then the standard deviation will become smaller. Conversely, if a score shifts away from the mean, then the standard deviation will increase. Third, like the mean, the standard deviation is stable with regard to sampling fluctuations. If samples were taken repeatedly from populations of the type usually encountered in the behavioral sciences, the standard deviation of the samples would vary much less from sample to sample than the range. This property is one of the main reasons why the standard deviation is used so much more often than the range for reporting variability. Variance The variance of a set of scores is just the square of the standard deviation or the average squared distance from the mean. FORMULA: 𝑠 2 = (𝑠)2 Where: 𝑠 2 = 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑠 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 Example: Computed Standard How to get the variance Variance Deviation 𝑠 2 = (3.56)2 𝑠 2 = 12.67 s = 3.56 𝑠 2 = (3.90)2 𝑠 2 = 15.21 s = 3.90 Characteristics of the Variance The variance is not used much in descriptive statistics because it gives us squared units of measurement. However, it is used quite frequently in inferential statistics. Reporting Standard Deviation In reporting the results of a study, the researcher often provides descriptive information for both central tendency and variability. The dependent variables in psychology research are often numerical values obtained from measurements on interval or ratio scales. With numerical scores the most common descriptive statistics are the mean (central tendency) and the standard deviation (variability), which are usually reported together. In many journals, especially those following APA style, the symbol SD is used for the sample standard deviation. For example, the results might state: Children who viewed the violent cartoon displayed more aggressive responses (M = 12.45, SD = 3.7) than those who viewed the control cartoon (M = 4.22, SD = 1.04). When reporting the descriptive measures for several groups, the findings may be summarized in a table. Refer to the table below for an example. B. MEASURES OF POSITION It determines the standing or position of a single value (score) in relation to other values or scores in a sample or population data set. Measures of position give us a way to see where a certain data point or value falls in a sample or distribution. A measure can tell us whether a value is about the average, or whether it’s unusually high or low. Measures of position are used for quantitative data that falls on some numerical scale. Sometimes, measures can be applied to ordinal variables— those variables that have an order, like first, second…fiftieth. Quartile A distribution of test scores or data can be divided into four parts such that 25% of the test scores occur in each quarter. Thus, the first quartile cuts off the lowest 25%, the second quartile cuts off the lowest 50%, and the third quartile cuts off the lowest 75%. (Note that the second quartile is also the median.). The quartiles/quartile points represent the dividing points between the four quarters in the distribution. There are three of them, respectively labeled Q1, Q2, and Q3. To differentiate, quartile refers to a specific point whereas quarter refers to an interval. An individual score may, for example, fall at the third quartile or in the third quarter (but not “in” the third quartile or “at” the third quarter).  Simply put, quartiles divide your data into quarters: the lowest quarter, two middle quarters, and a highest quarter. 𝑁 − 𝑐𝑓𝑏 FORMULA: 𝑄1 = 𝐿 + 𝑖 ( 4 ) 𝑓𝑚 2𝑁 − 𝑐𝑓𝑏 𝑄2 = 𝐿 + 𝑖 ( 4 ) 𝑓𝑚 3𝑁 − 𝑐𝑓𝑏 𝑄3 = 𝐿 + 𝑖 ( 4 ) 𝑓𝑚 𝑛 = 𝑡𝑜𝑡𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑖𝑒𝑠 𝐿 = 𝑟𝑒𝑎𝑙 𝑙𝑜𝑤𝑒𝑟 𝑙𝑖𝑚𝑖𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒 𝑐𝑙𝑎𝑠𝑠 Where: 𝑖 = 𝑐𝑙𝑎𝑠𝑠 𝑠𝑖𝑧𝑒/ 𝑐𝑙𝑎𝑠𝑠 𝑤𝑖𝑑𝑡ℎ 𝑐𝑓𝑏 = 𝑐𝑓 𝑏𝑒𝑙𝑜𝑤 𝑡ℎ𝑒 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒 𝑐𝑙𝑎𝑠𝑠 𝑓𝑚 = 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒 𝑐𝑙𝑎𝑠𝑠  Example: Compute for Quartiles 1, 2, and 3 Class Intervals f cf rf p cp 15-17 4 40 0.10 10% 100% 12-14 6 36 0.15 15% 90% 9-11 8 30 0.20 20% 75% 6-8 12 22 0.30 30% 55% 3-5 10 10 0.25 25% 25% i=3 ∑f = 40 (N) ∑rf = 1 ∑ 𝑝 = 100% Step 1: Compute for n by adding all frequencies Step 2: Fill in the cumulative frequency column starting from the lowest CI Step 3: Use one of the formulas appropriate to the quartile point you are to look for a. First, you must identify the quartile class using one of the ff formulas 𝑄𝑛 C.I./Quartile class fm 𝑐𝑓𝑏 𝑁 40 𝑄1 10 3-5 10 0 4 4 2𝑁 2(40) 𝑄2 20 6-8 12 10 4 4 3𝑁 3(40) 𝑄3 30 9-11 8 22 4 4 b. Proceed by substituting the values to the formula 𝑄𝑢𝑎𝑟𝑡𝑖𝑙𝑒1 𝑄𝑢𝑎𝑟𝑡𝑖𝑙𝑒2 𝑄𝑢𝑎𝑟𝑡𝑖𝑙𝑒3 𝑁 2𝑁 3𝑁 − 𝑐𝑓𝑏 − 𝑐𝑓𝑏 − 𝑐𝑓𝑏 𝑄1 = 𝐿 + 𝑖 ( 4 ) 𝑄2 = 𝐿 + 𝑖 ( 4 ) 𝑄3 = 𝐿 + 𝑖 ( 4 ) 𝑓𝑚 𝑓𝑚 𝑓𝑚 40 2(40) 3(40) −0 − 10 − 22 𝑄1 = 2.5 + 3 ( 4 ) 𝑄2 = 5.5 + 3 ( 4 ) 𝑄3 = 8.5 + 3 ( 4 ) 10 12 8 𝑄1 = 5.5 𝑄2 = 8 𝑄3 = 11.5 * 25% of the scores fall below * 50% of scores fall below * 75% of scores fall below 5.5 8 11.5 * 25% of the test takers * 50% of the test takers * 75% of the scores fall obtained a score below 5.5 obtained a score below 8 below 11.5  MEDIAN QUARTILE 2 𝑛 2𝑁 − 𝑓𝑐 − 𝑐𝑓𝑏 𝑀𝑑 = 𝐿 + 𝑖 (2 ) 𝑄2 = 𝐿 + 𝑖 ( 4 ) 𝑓𝑚 𝑓𝑚 40 2(40) − 10 − 10 𝑀𝑑 = 5.5 + 3 ( 2 ) 𝑄2 = 5.5 + 3 ( 4 ) 12 12 𝑀𝑑 = 8 𝑄2 = 8 Md = 𝑸𝟐 *Median is equal to Quartile 2 Decile Deciles are similar to quartiles except that they use points that mark 10% rather than 25% intervals. Thus, the top decile, or D9, is the point below which 90% of the cases fall. The next decile (D8) marks the point below which 80% of the cases fall, and so forth. Thus, deciles split the data into ten equal parts, with the first decile cutting off the lowest 10%, the second decile cutting off the lowest 20%, and so on. FORMULA: 𝑘(𝑛) − 𝑐𝑓𝑏 𝐷𝑘 = 𝐿 + 𝑖 ( 10 ) 𝑓𝑚 Where: 𝑛 = 𝑡𝑜𝑡𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑖𝑒𝑠 𝑘 = 𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝑑𝑒𝑐𝑖𝑙𝑒 𝑝𝑜𝑖𝑛𝑡/𝑑𝑒𝑐𝑖𝑙𝑒 𝑝𝑜𝑖𝑛𝑡 𝑡𝑜 𝑙𝑜𝑜𝑘 𝑓𝑜𝑟 𝐿 = 𝑟𝑒𝑎𝑙 𝑙𝑜𝑤𝑒𝑟 𝑙𝑖𝑚𝑖𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑒𝑐𝑖𝑙𝑒 𝑐𝑙𝑎𝑠𝑠 𝑖 = 𝑐𝑙𝑎𝑠𝑠 𝑠𝑖𝑧𝑒/ 𝑐𝑙𝑎𝑠𝑠 𝑤𝑖𝑑𝑡ℎ 𝑐𝑓𝑏 = 𝑐𝑓 𝑏𝑒𝑙𝑜𝑤 𝑡ℎ𝑒 𝑑𝑒𝑐𝑖𝑙𝑒 𝑐𝑙𝑎𝑠𝑠 𝑓𝑚 = 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑒𝑐𝑖𝑙𝑒 𝑐𝑙𝑎𝑠𝑠  Example: Compute for Deciles 3, 5, and 7 Class Intervals f cf rf p cp 15-17 4 40 0.10 10% 100% 12-14 6 36 0.15 15% 90% 9-11 8 30 0.20 20% 75% 6-8 12 22 0.30 30% 55% 3-5 10 10 0.25 25% 25% i=3 ∑f = 40 (N) ∑rf = 1 ∑ 𝑝 = 100% Step 1: Compute for n by adding all frequencies Step 2: Fill in the cumulative frequency column starting from the lowest CI Step 3: Use one of the formulas appropriate to the decile you are to look for a. First, you must identify the decile class using one of the ff formulas 𝐷𝑛 C.I./Decile class fm 𝑐𝑓𝑏 𝐷3 𝑘(𝑛) 3(40) 12 6-8 12 10 10 10 𝐷5 𝑘(𝑛) 5(40) 20 6-8 12 10 10 10 𝐷7 𝑘(𝑛) 7(40) 28 9-11 8 22 10 10 b. Proceed by substituting the values to the formula 𝐷𝑒𝑐𝑖𝑙𝑒 3 𝐷𝑒𝑐𝑖𝑙𝑒 5 𝐷𝑒𝑐𝑖𝑙𝑒 7 𝑘(𝑛) 𝑘(𝑛) 𝑘(𝑛) − 𝑐𝑓𝑏 − 𝑐𝑓𝑏 − 𝑐𝑓𝑏 𝐷3 = 𝐿 + 𝑖 ( 10 ) 𝐷5 = 𝐿 + 𝑖 ( 10 ) 𝐷7 = 𝐿 + 𝑖 ( 10 ) 𝑓𝑚 𝑓𝑚 𝑓𝑚 3(40) 5(40) 7(40) − 10 − 10 − 22 𝐷3 = 5.5 + 3 ( 10 ) 𝐷5 = 5.5 + 3 ( 10 ) 𝐷7 = 8.5 + 3 ( 10 ) 12 12 8 𝐷3 = 6 𝐷5 = 8 𝐷7 = 10.75 * 30% of the scores fall below * 50% of scores fall below * 70% of scores fall below 6 8 10.75 * 30% of the students got a * 50% of the students got * 70% of the students got score of below 6 a score of below 8 a score of below 10.75  MEDIAN QUARTILE 2 DECILE 5 𝑛 2𝑁 𝑘(𝑛) − 𝑓𝑐 − 𝑐𝑓𝑏 − 𝑐𝑓𝑏 𝑀𝑑 = 𝐿 + 𝑖 (2 ) 𝑄2 = 𝐿 + 𝑖 ( 4 ) 𝐷5 = 𝐿 + 𝑖 ( 10 ) 𝑓𝑚 𝑓𝑚 𝑓𝑚 40 2(40) 5(40) − 10 − 10 − 10 𝑀𝑑 = 5.5 + 3 ( 2 ) 𝑄2 = 5.5 + 3 ( 4 ) 𝐷5 = 5.5 + 3 ( 10 ) 12 12 12 𝑀𝑑 = 8 𝑄2 = 8 𝐷5 = 8 Md = 𝑸𝟐 = 𝑫𝟓 *Median is equal to Quartile 2 and Decile 5 Percentile A percentile is a ranking that conveys information about the relative position of a score within a distribution of scores. More formally defined, a percentile is an expression of the percentage of people whose score on a test or measure falls below a particular raw score. Percentile is used extensively in education to compare the performance of an individual to that of a reference group. Percentile is a measure that divides the distribution into 100 equal parts. Further, it is the value on the measurement scale below which a specified percentage of the scores in the distribution fall. For example, if 80th percentile (P80) = 65, then 80% of all examinees scored below 65. FORMULA: 𝑘(𝑛) − 𝑐𝑓𝑏 𝑃𝑘 = 𝐿 + 𝑖 ( 100 ) 𝑓𝑚 𝑛 = 𝑡𝑜𝑡𝑎𝑙 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑖𝑒𝑠 𝑘 = 𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 𝑝𝑜𝑖𝑛𝑡/𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 𝑝𝑜𝑖𝑛𝑡 𝑡𝑜 𝑙𝑜𝑜𝑘 𝑓𝑜𝑟 Where: 𝐿 = 𝑟𝑒𝑎𝑙 𝑙𝑜𝑤𝑒𝑟 𝑙𝑖𝑚𝑖𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 𝑐𝑙𝑎𝑠𝑠 𝑖 = 𝑐𝑙𝑎𝑠𝑠 𝑠𝑖𝑧𝑒/ 𝑐𝑙𝑎𝑠𝑠 𝑤𝑖𝑑𝑡ℎ 𝑐𝑓𝑏 = 𝑐𝑓 𝑏𝑒𝑙𝑜𝑤 𝑡ℎ𝑒 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 𝑐𝑙𝑎𝑠𝑠 𝑓𝑚 = 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 𝑐𝑙𝑎𝑠𝑠  Example: Compute for Percentiles 15, 50, and 95 Class Intervals f cf rf p cp 15-17 4 40 0.10 10% 100% 12-14 6 36 0.15 15% 90% 9-11 8 30 0.20 20% 75% 6-8 12 22 0.30 30% 55% 3-5 10 10 0.25 25% 25% I=3 ∑f = 40 (N) ∑rf = 1 ∑ 𝑝 = 100% Step 1: Compute for n by adding all frequencies Step 2: Fill in the cumulative frequency column starting from the lowest CI Step 3: Use one of the formulas appropriate to the percentile you are to look for a. First, you must identify the percentile class using one of the ff formulas 𝑃𝑛 C.I./Percentile Class fm 𝑐𝑓𝑏 𝑘(𝑛) 15(40) 𝑃15 6 3-5 10 0 100 100 𝑘(𝑛) 50(40) 𝑃50 20 6-8 12 10 100 100 𝑘(𝑛) 95(40) 𝑃95 38 15-17 4 36 100 100 b. Proceed by substituting the values to the formula 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 15 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 50 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑖𝑙𝑒 95 𝑘(𝑛) − 𝑐𝑓𝑏 𝑃15 = 𝐿 + 𝑖 ( 100 ) 𝑘(𝑛) − 𝑐𝑓𝑏 𝑘(𝑛) − 𝑐𝑓𝑏 𝑓𝑚 𝑃50 = 𝐿 + 𝑖 ( 100 ) 𝑃95 = 𝐿 + 𝑖 ( 100 ) 15(40) 𝑓𝑚 𝑓𝑚 −0 𝑃15 = 2.5 + 3 ( 100 ) 50(40) − 10 95(40) − 36 10 𝑃50 = 5.5 + 3 ( 100 ) 𝑃95 = 14.5 + 3 ( 100 ) 𝑃15 = 4.3 12 4 𝑃50 = 8 𝑃95 = 16 * 15% of the scores fall below 4.3 * 50% of scores fall below 8 * 95% of scores fall below 16 * 15% of the examinees * 50% of the examinees * 95% of the examinees scored below 4.3 scored below 8 scored below 16