HELP PLEAZEEEE MEASURES OF POSITION FOR UNGROUPED DATA please help it's so hard

Ill. Introduction Our lesson today is about measures of position. Measures of position, also known as quantiles, determines the position of single value in relation to other values or a population data set. There are three common measures of position, the quartiles, deciles, and percentiles. These are computed differently depending whether it is grouped or ungrouped data. Here is a summary how of we are going to go through the lessons. Quartile Ungrouped Data Decile Percentile Measures of Position Quartile Grouped Data Decile Percentile In this lesson, we are going to tackle about the Quartiles, Deciles and Percentile Ungrouped Data ALL YOU NEED TO KNOW! Quartiles are summary measures that divide a ranked (arranged from least to greatest) data set into four equal parts. Which is denoted by three data points below. 25% 25% 25% 25% Q2 Q3 first quartile middle quartile third quartile Q1 (first quartile) is called the lower quartile where 25% of the data has a value less than or equal to it. Q2 (second quartile) is called the middle quartile where 50% of the data has a value less than or equal to it. Q3 (third quartile) is called the upper quartile where 75% has a value less than or equal to it. Decile are summary of measures that divide a ranked data set into 10 equal parts with nine data points. They are denoted as D1, D2, D3, ..., Do 10% 10% 10% 10% 10% 10% 10% 10% 10% 10% D2 first second third D5 D6 D8 docile decile fourth fifth decile sixth seventh decile eighth lecile ninth decile decile decile decile D1 is the 1st decile which pertains to the 10% of the data distribution equal or below it D2 is the 2nd decile which pertains to the 20% of the data equal or below it D3 is the 3d decile which pertains to the 30% of data equal or below it ,..., and so on, Do is the 9th decile which pertains to the 90% of data equal or below it.Percentiles are the ninety-nine score points which divide a ranked data set into one hundred equal parts. It is used to characterized values according to the percentage below them. It is denoted as P1, P2, P3, ..., P100. 1% 1% 1% 1% 1% 1% 1% 1% 1% 1% 1% P99 The first percentile (P,) separates the lowest 1% from the other 99%, the second percentile (P2) separates the lowest 2% from the other 98%, and so on. The P, is the 1%, P2 is the 2%, P3 is the 3%, P4 is the 4%, ..., Pso is the 50%, ..., and Poo is the 99% below or equal to the data. The Median There are positions in quartile, decile and percentile that has the same value. For example, D2 is equal to P20 since both means 20% below or equal to the data set. Q3 is also equal to P75, which both means 75% of the data set. The median of the distribution in quartile, decile and percentile are all equal. DS P50 Q2 = D5 = P50 = 50% IV. Skill Development Let us look at the examples on how the measure of position for ungrouped data is solved. Mendenhall and Sincich Method. The method is used to determine the position of the relative position if it has decimal. Use the following rule in rounding of the th position. For quartile For first quartile (Q1), round up all the decimal values. For upper quartile (Q3), round down all the decimal values. Round Up Round Down Ex. 3.54 ~ 4 For Decile Ex. 12.78 = 12 Round decimal values to the nearest integer. ex. 3. 35 - 3, or 7.54 ~ 8 For Percentile Round decimal values to the nearest integer. ex. 3. 35 ~ 3, or 7.54 ~ 8 The formula below will help you locate the ith position of the data Quartile Decile i(n + 1) th Percentile QI = i(n + 1) th + D = 10 PI = i(n + 1) th 100 where i = positionGeneral Steps in Solving Step 1: Arrange Step 4: Interpret Step 2: Solve the Step 3: Locate the data in ith position using the position in the result. ascending order the arranged (smallest to the appropriate data. formula. largest). Quartile for ungrouped data Example 1: The following table lists the calories per 100 ml of 25 popular sodas. Find the lower, middle and upper quartile of the data. Calories, per 100mL 40 53 9 9 56 26 73 41 36 58 39 Solution: Step 1: Arrange the data in ascending order. (least to greatest) 26 32 36 36 41 42 42 45 50 53 53 (Total Number of data) n = 25 Lower Quartile Middle Quartile Upper Quartile Step 2: Solve the position of the Step 2: Solve the position of the Step 2: Solve the position of the lower lower quartile. lower quartile. quartile Qi = i(n+1) th * substitute value: Q1 = in+1)th * substitute value: 21 = in+1)th * substitute value: i = 1 and n = 25 i = 2 and n = 25 i = 3 and n = 25 Q1 = 1(25+1) *add then multiply Q2 = 2(25+1) *add then multiply Q3 = 3(25+1) *add then multiply Q1 = 26 * divide 22 = * divide 23 = * divide Q1 = 6.5 th *The rule for Q1 is to Q2 = 13 th *The answer is a Q3 = 19.5 th * The rule for Q3 is round up to the whole number. No to round down to ~ 7th nearest integer. need to round it. ~ 19th nearest integer. Step 3: Determine the value of the Step 3: Determine the value of the position. To do this is count up to 7th ith position. To do this is count up to Step 3: Determine the value of the ith place in the arranged data. 13th place in the arranged data. position. To do this is count up to 19th place in the arranged data, The 7th place on the arranged The 13th place on the arranged data is 39. Therefore, Q1 = 39. data is 43. Therefore, Q2 = 43. The 19th place on the arranged data is 50. Therefore, Q3 = 50. Step 4: Interpret the result. Step 4: Step 4: Interpret the Result This means that 25% of This means that 50% of calories per 100 mL of | the 25 calories per 100 mL of the 25 This means that 75% of popular sodas is below or equal popular sodas is below or equal calories per 100 mL of the 25 to 39 calories. Likewise, 75% of to 43 calories. Likewise, 50% of popular sodas is below or equal the calories per 100 mL of the 25 the calories per 100 mL of the 25 to 50 calories. Likewise, 75% of popular sodas is greater than or popular sodas is greater than the calories per 100 mL of the 25 equal to 39 calories. or equal to 43 calories. popular sodas is greater than or equal to 50 calories.Deciles for ungrouped data Example 2: The following are summative exam scores of fifteen random students in Grade 10. Find the 1 st decile and 6th decile of the data. 35, 42 40, 28, 15, 23, 33, 20, 18, 28, 35, 20, 41, 16, 32 Solution: Step 1: Arrange the data from smallest to largest (ascending order). 15, 16, 18, 20, 20, 23, 28, 28, 32, 33, 35, 35, 40, 41, 42 Step 2: Compute the location of the D, and De position, using the formula i(n + 1) th DI = Where, i = 1 * Since we are looking for D, Where, i = 6 *Since we are looking for D6 n = 15 * There are 15 given data. n = 15 *There are 15 given data. Compute: Compute: i(n+1)th * Substitute i and n Di = i(n+1)th Di = * Substitute i and n 10 10 D, 1(15+1) th Add, then multiply D =6(15+1) th * Add, then multiply LO D1 = 16th * Divide 96th D6 * Divide D1 = 1.6th * The rule for decile is to round to the nearest D6 = 9.6th * The rule for decile is to ~ and integer round to the nearest ~ 10th the nearest integer. Step 3: Locate the second position in the arranged data. Step 3: The value of the 2nd place in the arranged The 10th place in the arranged data is 33. data is 16. Therefore, D1 = 16. Therefore, the value of De is 33. Step 4: Interpret. Step 4: Therefore, 10% of the students have scores below or equal to 16 points in the This means that 60% of the student's summative exam. If Jenny has a score of score is below or equal to 33 points. exactly 16 points, then she surpassed 10% of her classmates in the exam, but 90% of her classmates outperformed her.Percentile for ungrouped data Example 3: For the following height data collected from 30 students, find the 10th and 95th percentiles. 91 , 89, 88, 87, 89, 91, 87, 92, 90, 98, 95, 97, 96, 100, 101, 96, 98, 99, 98, 100, 102, 99, 101, 105, 103, 107, 105, 106, 107, 112. Solution: Step 1: Arranged the data. 87, 87, 88, 89, 89, 90, 91, 91, 92, 95, 96, 96, 97, 98, 98, 98, 99, 99, 100, 100, 101, 101, 102, 103, 105, 105, 106, 107, 107, 112. n = 30 i(n+1) th Step 2: Solve for the 10th and 95th percentile location using the formula, P, = 100 For Pio. For P95. - i(n+1)th *Where i =10 and n = 30 I (n+1)th *Substitute i = 95 and n = 30 100 PI = 100 Pio = 10(30+1) th 95 (30+1) th * Add, then multiply 100 100 P10 = 310th 2945th P95 = * Divide 100 100 P10 = 3.1th * The rule of percentile is to Pos = 29.45th * The rule of percentile is to round to the nearest integer. round to the nearest integer. ~ 3rd ~ 29th The 3rd value of the sorted data is 88. The 29th value of the sorted data is 107. Therefore, P10 = 88. Therefore, Pos = 107. This means that 10% of the height of It means that 95% of the student's height is students is less than or equal to 88. below or equal to 107. Otherwise, 5% is greater than 107. If your height is on the 95th percentile, a common interpretation is that only 5% of the height were higher than yours. Linear Interpolation The other way to solve the position when the location has a decimal is to interpolate. Linear interpolation is the simplest method of getting the exact values at position between two points. Unlike the mendenhall and sincich method, the decimal will not be rounded off. To solve for interpolation, we use: where : Value = LV + [D (HV - LV)] lower value ZIO higher value decimal value in the position Example 4: Using the same data above, solve the 1st Quartile, 5th Decile and 75th Percentile of the height of 30 students using interpolation. 88, 98. 91, 92, 99. 92, 95, 105 96, 107. 101 . 96, 106. 107. 100, 100, 12 101, 102, 103,75th Percentile 1st Quartile 5th Decile Q1 = in+1)th P, In+1)th D, = In+1) th 100 10 Q1 = 1(30+1) th D= = 5(30+1) the P75 - 75(30+1) th 10 100 Q1 = 31th D, = 155ch 2325th 10 100 Q1 = 7.75th D = 15.5th P, = 23.25th This means the data is in the 23rd and This means the data is between This means the data is between the 15th and 16th place. 24th place. the 7th and 8th place. 15th place value = 98 23rd place value = 102 7th place value = 91 16th place value = 98 24th place value = 103 8th place value = 92 Decimal = 0.75 Decimal = 0.5 Decimal = 0.25 Interpolate: Interpolate: Interpolate: Lower value = 91 LV = 98 LV = 102 Higher value = 92 HV = 98 HV = 103 Value = LV + [D (HV - LV)] Value = LV + [D (HV - LV)] Value = LV + [D (HV - LV) ] Value = 91 + [0.75 (92 -91)] Value = 98 + [0.5 (98 -98)] Value = 102 + [0.25 (103 - 102)] Value = 91 + [0.75 (1)] Value = 98 + [0.5 (0)] Value = 102 + [0.25 (1)] Value = 91 + 0.75 Value = 98 + 0 Value = 102 + 0.25 Value = 91.75 Value = 98 Value = 102.25 Therefore, 25% of the Therefore, 50% of the Therefore, 75% of the height is less than or equal to 102.25. height is less than or equal to height is less than or equal to 91.75. 98. Note! If the ith position does not have a decimal, there is no need to interpolate. Other examples involving measures of position Example 5: The lower quartile of a data set is the 8th data value. How many data values are in the data set? Solution: Using the formula using the lower quartile, compute the number of data (n). 21 = 1(n+1) *Substitute the Q1 = 8 8 = 1(n+1) *Cross Multiply 32 = n + 1 * Solve 32 - 1 = n 31 = n Therefore, there are 31 values in the data set. Example 6: Mrs. Santos is working as an engineer in a big firm in Cebu City. Her salary is in the 7.5th decile. Should Mrs. Santos be glad about her salary? Why? Solution: The 7.5th decile in percentage (7.5 * 10) is equivalent to 75%. Seventy-five percent (75%) of the employees receive a salary less than or equal to her salary and 25% of the salary. employees receive a salary greater than Mrs. Santos' salary. She should be glad with herPercentile Rank for Ungrouped Data Example 7: If the scores of a set of students in a math test are 20, 30, 15 and 75, what is the percentile rank of the score 30? Arrange the score: Rank Score Rank of the x - x 100 Use the formula: Percentile of score x = = total number of data values (n) P = R (100) * Substitute Rank: R = 3 and n = 4 P = 3 (100) * Divide P = (0.75) 100 * Multiple P = 75 *Answer Therefore, the score 30 is in the 75th Percentile. V. Guided Practice A. Arthur has an assignment to ask at random 15 students in their school about their ages. The data are given in the table below. Determine the 3d Decile, 50th Percentile and 2nd Quartile of the data. Interpret the result. Use the Mendenhall and Sincich method. Name Age Name Hanabi Janeth Claud Alberto Eudora Panda 22 Laila Ciara Hannah Kira Marcus Pain Gladys Naruto Ronnie Step 1: Arrange the data. 23 Step 2: Solve the location or position of the D3, Pso, and Q2. Remember to use the appropriate formula. For D3 For P50 For Q2Step 3: Determine the value of the position: The D3 data is in the th place and the value of it in arrange data is The P50 data is in the th place and the value of it in arrange data is The Q2 data is in the th place and the value of it in arrange data is Step 4: Interpret the results for D3, Pso and Q2. Hence, B. The following data are the random age of 14 individuals who visited Carmen Plaza. Solve the 3d decile and middle quartile of the data below using linear interpolation. Show your solution. 35, 42, 40, 28, 15, 23, 33, 20 18, 25, 27, 13, 50, and 45 C. If the scores of a set of students in a math test are 20, 30, 15, 75, 45, 50 what is the percentile rank of the score 45? Step 1: Arrange the score: Rank Score Step 2: Use the formula: Percentile = Rank of the x total number of data values (n) X (100) Where: Rank = and n = Solve: Therefore,VI. Relevance of the lesson Solving the measures of position of the data allows you to decide the position of the individual against the population. It means the rank of an individual based on the population of people, not on the standard score. It facilitates categorization of data sets and observations into samples for convenience of analysis and measurement. You can use QUARTILES to find the top 25 percent of incomes in a population or below 25 percent of the population. You can use DECILES if you want when you want the data set divided into ten equal parts. For example, Australia uses decile ranks to report drought data. Deciles 1-2 represent the lowest 20% (below normal). That means droughts that are "much below normal" don't occur more than 20% of the time. The PERCENTILES scores are frequently used to report results from national standardized test such as National Achievement Test (NAT) and the National Career Assessment Examination (NCAE). It shows the percentage of scores that a particular score surpassed. For example, if your score is 75 points on a test, and are ranked in the 85th percentile, it means that your score is higher than the 85% of the scores of the students who took the test. In summary, measure of position shows how the values from different distributions or measurement scales compare. A measure can tell us whether a value is about the average, or whether it's high or low. VII. Practice Exercise Show your skills! Encircle the letter of your desired answer. 1. The lower quartile is interpreted as the of the distribution. a. 25% b. 50% C. 75% d. 100% 2. When the distribution is divided into 4 equal parts, each score point that describes the distribution is called a a. Percentile b. Decile c. Quartile d. Median For questions 3, 4 and 5, use the data in the table below. The following are the scores of 14 students in History exam. 3. What is the lower quartile of the score in History exam? Interpolate if necessary. a. 3.75 b. 27 C. 14 d. 28.5 4. Based on the data above, what is the position of the lower quartile? a. 3.75th place b. 27th place c. 14th place d. 28.5th place 5. Compute the Q2 of the data? What does this mean? a. Fifty percent of the student scores are below or equal to 36.5. Seventy-five percent of student scores is equal to 36.5 The students score is above 38 points. d. The students score is below the passing rate which is 37. data set? 6. The upper quartile of a data set is the 15th data value. How many data values are in the a. 18 b. 15 C. 19 d. 207. The 4th decile is interpreted as Forty percent below or equal of the data position. Forty-five percent below or equal of the data position. Forty percent above of the data position. The median of the data position. 8. The Dso , is equivalent to the: d. 6th Decile 1 st Quartile b. 2nd Quartile C. 3rd Quartile 9. What is the seventh decile from the following array of data? Interpolate the result. 20, 28, 29, 30, 36, 37, 39, 42, 53, 54 d. 42.1 a. 41.1 b. 30.5 C. 39 10. Compute the 5th decile of the data: 5, 3, 7, 9, 6. b. 17 d. 9 11. If the score of the students is at the 6th decile or Deo in a 60-item test. Which of the following statement is NOT true? a. His score is below or equal to the 60% of the students. There are 40% of the students who did better than him. The passing percentage of the quiz is 60 points. d. All the statement are incorrect. 12. Percentile rank refers to the percentage of scores that fall above a certain score the percentage of scores that fall at or above a certain score the percentage of scores that equal a certain score d. the percentage of scores that fall at or below a certain score 13. The 75th percentile is also known as the a. median b. third quartile C. 1 st quartile d. interquartile range 14. Given the data, calculate the 80th Percentile. 9, 10, 12, 13, 14, 15, 15, 16, 17, 17, 18, 18, 19, 20, 21, 22, 25, 27, 29, 30 a. 21 b. 22 C. 25 d. 27 15. What is the 50th percentile of the following student heights: 60 in, 75 in, 70 in, 58 in, 62 S a. 58 b. 70 C. 75 d. 62 16. For the data set below, 102 is in what percentile? Round to nearest integer. 98, 99, 99, 100, 101, 102, 104, 104, 105, 105, 107, 110, 112, 112 a. 57th b. 43rd C. 40th d. 42nd