mathematics as a tool major in Elementary Education

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1. Mathematics as a Tool Chapter 1. Data Management RHEA C. GATON, MAT Course Facilitator

2. DATA MANAGEMENT

3. Gathering and Organizing of Data The data are the quantities (numbers) or qualities (attributes) measured or observed that are to be collected and/or analyzed. Categorical data are nominal (gender and civil status) and ordinal scales (education and income level) Continuous data are interval (age and Celsius and ratio scales (height and income)

4. Variable - A characteristic or property that can take on different values for different individuals or items in a population or sample. Independent Variable: the variable being manipulated. It's considered the "cause" or "input" in an experiment. It is not affected by other variables in the experiment. Dependent Variable : measured in the experiment. It's considered the "effect" or "output" because it depends on changes made to the independent variable.

5. Quantitative Variable: Can be measured numerically (e.g., height, weight). Qualitative Variable : Describes a category or quality (e.g., gender, type of car).

6. Data gathered can be presented in textual, tabular, graphical or a combination of these. Textual Presentation uses statements with numerals in order to describe and interpret the data. Tabular Presentation uses statistical table to directly display the quantities or values collected as data. Graphical Presentation illustrates data in a form of graphs. Examples: bar graph, pie chart and line graph.

7. The data gathered should be properly organized in to grouped data called frequency distribution. Frequency Distribution - A table or chart that shows the number of times (frequency) each data value or range of values occurs. Steps : Determine as to estimate number of classes k, k = 1 + 3 log n , where n is the number of population. Determine the range, r = highest value - lowest value Obtain the class size, c = range\k Set the lowest value as the first lower limit and get the upper limit which equal to first lower limit + class size - 1 Do the same process again until you reach the last class limit that includes the highest value from the data.

8. Example 1. Construct a frequency table for the following data: 11 19 11 15 16 10 16 16 15 17 10 27 21 11 13 21 10 16 11 19 24 12 22 13 19 13 18 20 21 11 19 15 11 25 29 23 16 23 10 17 11 27 16 24 12 21 13 12 26 15 11 14 10 12 11 15 18 12 20 13

9. Example 2. Construct a frequency table for the scores of students in a Geometry test. 55 63 44 37 50 57 44 57 42 46 58 40 54 65 39 27 28 56 38 45 30 35 56 78 55 27 50 28 44 58 39 37 65 43 33 70 60 61 60 44

10. Interpretation of Data Any given data in statistics are useless if not interpreted. Descriptive statistics - refers to the process of using summary measures to describe the main features of a data set. These statistics are used to provide a simple overview of the sample and its characteristics, often by representing it in tables, graphs, or summary values. Descriptive statistics can be divided into:

11. Measures of central tendency - These summarize the center of a data set and include: Mean, Median, and Mode. Measures of variability (or dispersion) - These describe how spread out the data is and include: Range, Variance, Standard deviation. Measures of distribution shape - These describe the overall pattern of the data, including: Skewness and Kurtosis

12. Central Tendency - A measure that represents the center or typical value of a data set. The most common measures are: Mean : The arithmetic average. Median : The middle value of an ordered data set. Mode : The most frequently occurring value.

13. Measures of Central Tendency of Ungrouped Data 1. MEAN (Average) The mean is the sum of all values in a data set divided by the number of values. It represents the "average" and is often used when data is evenly distributed.

14. 2. MEDIAN The median is the middle value in an ordered data set. If there’s an odd number of values, it’s the center value. If even, it’s the average of the two middle values. Steps to Find the Median : Order the data from smallest to largest. Identify the middle value. If there’s an even number of values, take the average of the two central numbers.

15. 3. MODE The mode is the value that appears most frequently in a data set. There can be more than one mode (bimodal, multimodal), or none at all if all values are unique. Example : In the set {2, 4, 4, 5, 6, 6, 6, 8}, the mode is 6

16. WEIGHTED MEAN The weighted mean (or weighted average) is a type of mean where each value in a data set is multiplied by a weight that reflects its importance or frequency. Unlike a simple mean, which treats all values equally, the weighted mean gives more influence to some values based on their assigned weights. When to Use the Weighted Mean Unequal Importance : When values in a data set have different levels of importance or frequency (e.g., grades in courses with different credit hours). Data with Frequencies : When you have grouped data or values that occur with different frequencies

17.   FORMULA FOR WEIGHTED MEAN Where :  

18. EXAMPLE 1. There are 1,000 notebooks sold at Php 10.00 each; 500 notebooks at Php 20.00; 500 notebooks at Php 25.00, and 100 notebooks at Php 30.00. Compute the weighted mean

19. EXAMPLE 2. A teacher calculates a class average based on test scores from different sections, each with a different number of students: Section 1: Average score = 75, with 10 students; Section 2: Average score = 80, with 15 students; Section 3: Average score = 90, with 5 students

20. EXAMPLE 3. A supermarket stocks three categories of products with different prices and sales volumes: Category A: Average price per item = P150.00, sold 200 items Category B: Average price per item = P250, sold 120 items Category C: Average price per item = P400, sold 60 items Calculate the weighted average price per item across all categories.

21. ASSIGNMENT: There are 350 shirts sold at Php 100.00 each; 250 shirts at Php 150.00; 150 shirts at Php 200.00, and 65 shirts at Php 250.00. Compute the weighted mean.

22.   MEAN for GROUPED DATA Where :  

23. Steps to find the mean of grouped data: Find the class mark for each class interval:     2. Multiply each class mark by the class frequency:   3. Find the   4. Sum up all frequencies:   5. Divide: by to get the mean.  

24. EXAMPLE 1. The table below summarizes the weights of goats. Find the average weight of the goats . WEIGHT OF GOATS 201 – 210 3 191 – 200 8 181 – 190 12 171 – 180 11 161 – 170 9 151 – 160 2 WEIGHT OF GOATS 201 – 210 3 191 – 200 8 181 – 190 12 171 – 180 11 161 – 170 9 151 – 160 2

25. EXAMPLE 2. The table below shows the distribution of workers’ ages: WORKERS’ AGES 21 – 30 7 31 – 40 8 41 – 50 5 51 – 60 3 61 – 70 2 WORKERS’ AGES 21 – 30 7 31 – 40 8 41 – 50 5 51 – 60 3 61 – 70 2

26. EXAMPLE 3. The following are the scores of the students in an Algebra test. Make a frequency table and solve for the mean. 11, 14, 19, 21, 15, 24, 17, 20, 18, 23, 25, 20, 16, 22, 18, 19, 14, 21, 20, 15, 13, 17, 22, 23, 19, 18, 16, 17, 20, 24, 21, 15, 14, 19, 13, 18, 17, 20, 22, 16

27.   MEDIAN for GROUPED DATA Where:   NOTE: To determine the median class:  

28. CLASS INTERVAL FREQUENCY 28 – 29 1 60 26 – 27 3 59 24 – 25 3 56 22 – 23 3 53 20 – 21 6 50 18 – 19 6 44 16 – 17 8 38 14 – 15 6 = 30 median class 12 – 13 10 24 = 10 – 11 14 14 N = 60 CLASS INTERVAL FREQUENCY 28 – 29 1 60 26 – 27 3 59 24 – 25 3 56 22 – 23 3 53 20 – 21 6 50 18 – 19 6 44 16 – 17 8 38 14 – 15 30 median class 12 – 13 10 10 – 11 14 14 N = 60 EXAMPLE 1. Find the median of the following data:    

29. EXAMPLE 2. Find the median of the following data: < cf 75 – 79 6 60 70 – 74 7 54 65 – 69 2 47 60 – 64 8 45 55 – 59 1 2 37 50 – 54 7 25 45 – 49 1 18 40 – 44 8 8 N = 60 < cf 75 – 79 6 60 70 – 74 7 54 65 – 69 2 47 60 – 64 8 45 55 – 59 1 2 37 50 – 54 7 25 45 – 49 1 18 40 – 44 8 8 N = 60    

30.   MODE for GROUPED DATA Where:   NOTE: The modal class is the class with the highest frequency

31. CLASS INTERVAL FREQUENCY 28 – 29 1 26 – 27 3 24 – 25 3 22 – 23 3 20 – 21 6 18 – 19 6 16 – 17 8 14 – 15 6 12 – 13 10 10 – 11 14 modal class N = 60 EXAMPLE 1. Find the mode of the following data:    

32. Scores in Algebra FREQUENCY 75 – 79 6 70 – 74 7 65 – 69 2 60 – 64 8 55 – 59 1 2 50 – 54 7 45 – 49 1 40 – 44 8 N = 60 EXAMPLE 2. Find the mode of the following data:    

33. Scores in Algebra FREQUENCY 19 – 21 7 16 – 18 19 13 – 15 14 10 – 12 8 7 – 9 2 N = 50 Find the median and mode of the grouped data.

34. Measure of Relative Position The measure of relative position helps us understand how a particular data point compares to the other values in a dataset. Commonly used measures include percentiles, quartiles and deciles . These measures allow us to see how high or low a value is in relation to the rest of the data.

35. 1. Percentiles Percentiles are used to describe the position of a value relative to the entire dataset. They tell you what percentage of data values fall below a certain point. Definition : A percentile divides a dataset into 100 equal parts. Example : If you are at the 75th percentile, this means you scored better than 75% of the people

36. Ungrouped Data Examples: Mrs.Corpuz conducted a quiz to ten students. The scores obtained are as follows: 5, 8, 7, 6, 3, 6, 10,5,6,4 What score corresponds to the 100th percentile? What is the 50th percentile point? Solution: Arrange the scores in descending order. 10, 8, 7, 6,6, 6,5,5,4,3 The highest is 10, the middle is 6, and the lowest is 3. The one who scored 10 surpassed all the others. However, the class intervals will always have the upper boundary, so the 100th percentile point is the upper boundary of the highest score. P 10 = 10.5 b. Since the middle score is 6, it surpasses half (50%) of the students. Therefore, P 50 = 6

37. In a class of 50, Jason got a percentile rank of 65. What does this percentile rank imply? How many students rank below Jason? Solution: The P 65 implies that Jason got a score higher than 65 percent of the class. Since there are 50 students in all, the number of students who got scores below Jason is 50(65%) = 50(0.65) = 32.5

38.   PERCENTILE for GROUPED DATA Where:   NOTE: To determine the percentile class:  

39. CLASS INTERVAL FREQUENCY 28 – 29 1 60 26 – 27 3 59 24 – 25 3 56 22 – 23 3 53 20 – 21 6 50 18 – 19 6 44 16 – 17 8 38 14 – 15 6 30 12 – 13 10 24 10 – 11 14 14 N = 60 CLASS INTERVAL FREQUENCY 28 – 29 1 60 26 – 27 3 59 24 – 25 3 56 22 – 23 3 53 20 – 21 6 50 18 – 19 6 44 16 – 17 8 38 14 – 15 6 30 12 – 13 10 24 10 – 11 14 14 N = 60 EXAMPLE 1. Find the      

40. Scores in Algebra FREQUENCY 75 – 79 6 70 – 74 7 65 – 69 2 60 – 64 8 55 – 59 1 2 50 – 54 7 45 – 49 1 40 – 44 8 N = 60 EXAMPLE 2. Find the      

41. 2. Quartiles Quartiles divide a dataset into four equal parts. There are three main quartiles: Q 1 (First Quartile) : The 25th percentile, or the value that separates the lowest 25% of data. Q 2 (Second Quartile) : The 50th percentile, or the median of the dataset. Q 3 (Third Quartile) : The 75th percentile, or the value that separates the top 25% of data.

42.   QUARTILES for GROUPED DATA Where:   NOTE: To determine the quartile class: ; ;  

43. Scores in Algebra FREQUENCY 75 – 79 6 60 70 – 74 7 54 65 – 69 2 47 60 – 64 8 45 55 – 59 1 2 37 50 – 54 7 25 45 – 49 1 18 40 – 44 8 8 N = 60 Scores in Algebra FREQUENCY 75 – 79 6 60 70 – 74 7 54 65 – 69 2 47 60 – 64 8 45 55 – 59 1 2 37 50 – 54 7 25 45 – 49 1 18 40 – 44 8 8 N = 60 EXAMPLE 1. Compute        

45. Scores in Algebra FREQUENCY 75 – 79 6 60 70 – 74 7 54 65 – 69 2 47 60 – 64 8 45 55 – 59 1 2 37 50 – 54 7 25 45 – 49 1 18 40 – 44 8 8 N = 60 Scores in Algebra FREQUENCY 75 – 79 6 60 70 – 74 7 54 65 – 69 2 47 60 – 64 8 45 55 – 59 1 2 37 50 – 54 7 25 45 – 49 1 18 40 – 44 8 8 N = 60 EXAMPLE 2. Compute        

47. Scores in Algebra FREQUENCY 75 – 79 6 60 70 – 74 7 54 65 – 69 2 47 60 – 64 8 45 55 – 59 1 2 37 50 – 54 7 25 45 – 49 1 18 40 – 44 8 8 N = 60 Scores in Algebra FREQUENCY 75 – 79 6 60 70 – 74 7 54 65 – 69 2 47 60 – 64 8 45 55 – 59 1 2 37 50 – 54 7 25 45 – 49 1 18 40 – 44 8 8 N = 60 EXAMPLE 1.      

49. 3. Deciles Deciles are points that divide a distribution into ten equal parts. Each part is called a decile. So, , , …  

50.   DECILES for GROUPED DATA Where:   NOTE: To determine the decile class: ; ;  

51. Scores in Algebra FREQUENCY 75 – 79 6 60 70 – 74 7 54 65 – 69 2 47 60 – 64 8 45 55 – 59 1 2 37 50 – 54 7 25 45 – 49 1 18 40 – 44 8 8 N = 60 Scores in Algebra FREQUENCY 75 – 79 6 60 70 – 74 7 54 65 – 69 2 47 60 – 64 8 45 55 – 59 1 2 37 50 – 54 7 25 45 – 49 1 18 40 – 44 8 8 N = 60 EXAMPLE 2. Compute        

53. Scores in Algebra FREQUENCY 75 – 79 6 60 70 – 74 7 54 65 – 69 2 47 60 – 64 8 45 55 – 59 1 2 37 50 – 54 7 25 45 – 49 1 18 40 – 44 8 8 N = 60 Scores in Algebra FREQUENCY 75 – 79 6 60 70 – 74 7 54 65 – 69 2 47 60 – 64 8 45 55 – 59 1 2 37 50 – 54 7 25 45 – 49 1 18 40 – 44 8 8 N = 60 EXAMPLE 2. Compute      

55. Scores in Algebra FREQUENCY 75 – 79 6 60 70 – 74 7 54 65 – 69 2 47 60 – 64 8 45 55 – 59 1 2 37 50 – 54 7 25 45 – 49 1 18 40 – 44 8 8 N = 60 Scores in Algebra FREQUENCY 75 – 79 6 60 70 – 74 7 54 65 – 69 2 47 60 – 64 8 45 55 – 59 1 2 37 50 – 54 7 25 45 – 49 1 18 40 – 44 8 8 N = 60 EXAMPLE 2. Compute      

57. Measure of Variation Measures of variation describe the spread or dispersion of a dataset. They indicate how much the data values differ from each other or from the central tendency. The key measures of variation include range , variance , standard deviation , mean deviation , quartile deviation and interquartile range (IQR) .

58. 1. Range The simplest measure of variation: Ungrouped Data: Grouped Data:  

59. Scores in Algebra FREQUENCY 75 – 79 6 70 – 74 7 65 – 69 2 60 – 64 8 55 – 59 1 2 50 – 54 7 45 – 49 1 40 – 44 8 N = 60 EXAMPLE 1. Determine the range.      

60. 2. Mean Deviation The mean deviation is a measure of variation that makes use of all the scores in a distribution. This is more reliable than the range and quartile deviation. Ungrouped Data: Where:  

61. EXAMPLE 1. Find the mean deviation of the following ungrouped distribution: 4 , 8, 12. SOLUTION: a. Calculate for the mean   b. Complete the table: b. Substitute:   X 4 4 8 12 4 X 4 4 8 12 4

62.   MEAN DEVIATION for GROUPED DATA Where:  

63. X f 30 – 34 4 32 128 25 – 29 5 27 135 20 – 24 6 22 132 15 – 19 2 17 34 10 – 14 3 12 36 N = 20 X f 30 – 34 4 32 128 25 – 29 5 27 135 20 – 24 6 22 132 15 – 19 2 17 34 10 – 14 3 12 36 N = 20 EXAMPLE 1. Find the MD of the following. SOLUTION: a. Calculate for the mean of grouped data.  

64. X f 30 – 34 4 32 128 8.75 35 25 – 29 5 27 135 3.75 18.75 20 – 24 6 22 132 1.25 7.50 15 – 19 2 17 34 6.25 12.50 10 – 14 3 12 36 11.25 33.75 N = 20 X f 30 – 34 4 32 128 8.75 35 25 – 29 5 27 135 3.75 18.75 20 – 24 6 22 132 1.25 7.50 15 – 19 2 17 34 6.25 12.50 10 – 14 3 12 36 11.25 33.75 N = 20 b . Add columns and        

65. 2. Variance and Standard Deviation The standard deviation , SD is the most important and useful measure of variation. It is the square root of the variance , SD 2 It is an average to the degree to which each set of scores in the distribution deviates from the mean value. It is a more stable measure of variance because it involves all the scores in a distribution rather than the range. The standard deviation is the square root of the variance , showing the dispersion in the same units as the data.

66. Variance for Ungrouped Data   STEPS: Calculate the mean. Get the difference between each score and the mean, then square the difference. Get the sum of the squared deviation in step b. Substitute in the formula.

67. EXAMPLE 1. Find the variance and standard deviation of the following ungrouped distribution: 4 , 8, 12. SOLUTION: a. Calculate for the mean   b. Complete the table: c. Substitute:   X 4 -4 16 8 12 4 16 X 4 -4 16 8 12 4 16

68. C. Substitute: Therefore, the variance is 16.   STANDARD DEVIATION FOR UNGROUPED DATA The standard deviation is the square root of the variance Therefore, the standard deviation is 4.  

69. Variance for Grouped Data   STEPS: Calculate the mean. Get the difference between class mark and the mean, then square the difference. Find the product of the squared difference in step c and the frequency Get the sum of the squared deviation in step c. Substitute in the formula.

70. Population - The entire group or set of items or individuals that you want to study or make inferences about. For example, all students in a school. Sample - A subset of the population that is selected for analysis. For example, 100 students from the school.

71. Percentile - A measure that indicates the value below which a given percentage of observations fall. For example, the 90th percentile is the value below which 90% of the data lies. Quartiles - Values that divide a data set into four equal parts. - Q1 (First Quartile): The 25th percentile (25% of the data is below this value). - Q2 (Second Quartile): The median or 50th percentile. - Q3 (Third Quartile): The 75th percentile. Interquartile Range (IQR) - The difference between the third quartile (Q3) and the first quartile (Q1). It measures the spread of the middle 50% of the data.

72. Range - The difference between the highest and lowest values in a data set. Variance - A measure of how much the values in a data set differ from the mean. It quantifies the degree of variation or dispersion. Standard Deviation - The square root of the variance. It indicates the average distance of each data point from the mean and shows how spread out the data is.

73. Skewness - A measure of the asymmetry of the distribution of values in a data set. - Positive Skew (Right Skew): The tail on the right side is longer or fatter. - Negative Skew (Left Skew): The tail on the left side is longer or fatter. Kurtosis - A measure of the "tailedness" of the data distribution. It indicates whether the data is more or less outlier-prone than a normal distribution. - Leptokurtic: More peaked than a normal distribution. - Platykurtic: Flatter than a normal distribution. Outliers - Data points that are significantly different from the rest of the data. They can influence the mean and skew the results.

74. Histogram - A graphical representation of the frequency distribution of numerical data. The data is divided into intervals (bins), and the height of each bar represents the frequency of values in that data interval. Box Plot (Box-and-Whisker Plot) - A visual summary of da the median, quartiles, and potential outliers. It highlights the central tendency and variability in a data set.