Mathematics-as-a-Tool PDF

Mathematical as a Tool GE 4: Mathematics in the Modern World Prepared by: Tom Paulie Tongol  PRELIMINARY CONCEPTS The following terms are used frequently in this chapter: Statistics – refers to a set...

Mathematical as a Tool GE 4: Mathematics in the Modern World Prepared by: Tom Paulie Tongol  PRELIMINARY CONCEPTS The following terms are used frequently in this chapter: Statistics – refers to a set of mathematical procedures for organizing, summarizing, and interpreting information to draw a conclusion. Data – measurements or observations, raw information, and individual pieces of factual information recorded and used for the purpose of analysis.  Primary Data – refers to the first hand data gathered by the researcher himself. Example: national census data collected by the government Secondary Data – refers to the data collected by someone else. Example: data collected from online sources  Population – set of all the individuals of interest in a particular study. Sample – set of individuals selected from a population, usually intended to represent the population in a research study. Variable – a characteristic or condition that changes or has different values for different individuals.  QUALITATIVE VARIABLE – can be placed into distinct categories; categorial variable not placed on a meaningful number scale. (“not numerical”, “cannot be ordered/ranked”) QUANTITATIVE VARIABLE – one that is measurable using a meaningful number scale. (“numerical”, “can be ordered/ranked”) Types of Quantitative Variables Discrete Variables – values are obtained by counting. Example: number of children Continuous Variable – values are obtained by measuring. Example: temperature, distance Variables and Measurement Levels of Measurement Nominal – refers to the attribute of subjects or objects that is used for naming, labeling, and categorizing only. It cannot be arranged in an ordering manner nor computed using the fundamental operations (if numbers) for it serves only for counting or presentation purposes. Example: Id Number, Gender, Nationality, Car brands, Civil Status Levels of Measurement Ordinal Numbers – refers to the attribute of subjects or objects that is used for ranking and ordering. Example: Year Level, Level of satisfaction, Socio-economic status, educational qualification, Grades (assigned letters/numbers) Levels of Measurement Interval Numbers – refers to the attribute of subjects or objects that is of known in terms of sizes or distances. The number zero does not mean an absence of the characteristic or attribute of the subject or object. Interval scales hold no absolute zero (zero is arbitrary) and can represent values below zero. An interval scale means that the difference between numbers is the same. Example: Temperature 0° (Celsius & Fahrenheit), test scores Levels of Measurement Ratio Numbers – refers to the attribute of subjects or objects that contains the characteristic of the interval number but in this case, zero has a true value. The number zero indicates an absence of the characteristic or attribute of the subject or object. This is also called as the strongest level of measurement. Ratio scales differ from interval by having a character of origin, which is the starting or zero-point of the scale (zero is absolute). Example: Height, Weight, Monetary savings, Actual grades, Age Measures of Central Tendency Mean, Median, & Mode Measures of Central Tendency It is a single value that is used to identify the center of the data. It tends to lie within the center if it is arranged from lowest to highest and vice versa. There are three types of measures of central tendency namely: Mean, Median, and Mode. Mean Refers to the arithmetic average. Used when the data are in interval or ratio level of measurement. Used when the frequency distribution is regular, symmetrical or normal. Easily affected by extreme scores. Measures stability – “most stable”. Used to compare other measures such as standard deviation, coefficient of variation, skewness and z-score. Mean Also called as “arithmetic mean”, denoted as. It is the sum of all values in a data set divided by the number of values that are summed. It is written mathematically as: where x = individual value n = total number of values EXAMPLE #1: The following are the scores of the students on the GE 4 final exam. Calculate the mean? 42 47 39 57 55 31 46 24 18 23 SOLUTION: n = 10 FORMULA: 42 + 47 + 39 + 57 + 55 + 31 + 46 + 24 + 18 + 23 = 10 = 38.2 EXAMPLE #2: The following are the grades of Timmy for the 1st quarter: English = 93, Math = 89, Science = 91, Values = 76. What is his average for the 1st quarter? SOLUTION: n=4 FORMULA: 93 + 89 + 91 + 76 = 4 = 87.25 Median Refers to the centermost scores when the scores in the distribution are arranged according to magnitude (from highest to lowest score of vice versa). Used when the middlemost score is desired. Used when the data are in ordinal level of measurement. Used when the frequency distribution is irregular or skewed. Not affected by extreme scores because it is a positional measure. Most reliable measures of central tendency. EXAMPLE #1: Find the median of the following set of measurements. 61 28 58 76 16 50 65 25 39 SOLUTION: Arrange the data first in ascending order. 16 25 28 39 50 58 61 65 76 = 50 EXAMPLE #2: Find the median of the given data set. 3.3 6.6 3.0 9.8 3.7 2.9 5.5 8.0 SOLUTION: Arrange it in ascending order. 2.9 3.0 3.3 3.7 5.5 6.6 8.0 9.8 3.7+5.5 = = 4.6 2 Mode (Peak) Refers to the score/s that occurs most frequently in the score distribution. Used when the data is in nominal level of measurement. May not be unique. Not affected by extreme scores. May not exist at times. Types of Mode 1.) Unimodal – is a score distribution that consists of one mode. 2.) Bimodal – is a score distribution that consists of two modes. 3.) Trimodal – is a score distribution that consists of three modes and also considered as multi-modal that is consists of more than two modes. Types of Mode EXAMPLE #1: Find the mode of the following data set. 11 14 12 11 15 16 18 11 10 17 = 11 EXAMPLE #2: Find the mode of the following data sets. 2.6 4.2 3.5 2.6 4.2 3.6 2.1 4.9 4.2 2.6 = 2.6 & 4.2 EXAMPLE #3: Find the mode of the following data sets. 105 200 159 110 225 170 115 250 285 190 Since there is no value that occurs most frequently, then the mode is zero. = 0 Measures of Dispersion/Variability Range, Standard Deviation, & Variance Measures of Dispersion/Variability Measures of variability is used to find the spreadness of the data. The bigger the measure, the scatter is the data. A low dispersion indicates that the data points tend to be clustered tightly around the center. High dispersion signifies that they tend to fall further away. These are the types variability: Range, Standard Deviation, and Variance. Range The most unstable and unreliable measure because it can easily be affected by the extreme values. It is the difference of the highest and the lowest values in the distribution. FORMULA: R = H - L EXAMPLE: The following are the scores of the students on the GE 4 final exam. What is the range of this score distribution? 42 47 39 57 55 31 46 24 18 23 SOLUTION: Highest = 57; Lowest = 18 FORMULA: R = H - L R = 57 - 18 R = 39 Interpretation of Result for Range: Small Range: closer, clustered, homogenous, scores are less varied. (LEPTUKURTIC) Large Range: dispersed, scattered, spread apart, far from each other, heterogeneous, scores more varied. (PLATYKURTIC) KURTOSIS - the sharpness of the peak of a frequency-distribution curve. Standard Deviation (sd) Denoted as 𝝈. The most reliable measure of variability. The square root of variance. Standard deviation looks at how spread out a group of numbers is from the mean, by looking at the square root of the variance. How to calculate the sd? 1.) Find the mean. (x̄) 2.) Subtract the mean from each score to get the deviation. (d= x - x̄) 3.) Square the deviation. (d²) 4.) Get the sum of the squared deviations. (∑d²) 5.) Proceed to this formula to get the value of SD. σ 𝒅𝟐 𝝈= 𝑵−𝟏 N = total no. of scores EXAMPLE: x (d= x - x̄) d² 5 -4 16 7 -2 4 9 0 0 11 2 4 13 4 16 x̄ = 9 ∑d² = 40 σ 𝒅𝟐 𝟒𝟎 𝟒𝟎 𝝈= = = = 𝟏𝟎 = 𝟑. 𝟏𝟔 𝑵−𝟏 𝟓−𝟏 𝟒 Interpretation of Result of SD: Small SD: closer, clustered, homogeneous, scores are less varied. Large SD: dispersed, scattered, spread apart, far from each other, heterogeneous, scores are more varied. Variance (𝝈 ) 𝟐 Measures the dispersion of a set of data points around their mean. The variance is the average squared deviations from the mean. σ 𝟐 𝒅 𝐅𝐎𝐑𝐌𝐔𝐋𝐀: 𝝈𝟐 = 𝑵−𝟏 EXAMPLE: x (d= x - x̄) d² 5 -4 16 7 -2 4 9 0 0 11 2 4 13 4 16 x̄ = 9 ∑d² = 40 σ 𝟐 𝟐 𝒅 𝟒𝟎 𝟒𝟎 𝝈 = = = = 𝟏𝟎 𝑵−𝟏 𝟓−𝟏 𝟒 Measures of Relative Position PERCENTILES, QUARTILES, Z-SCORE Percentiles Divides the data into 100 equal parts. A student’s scores are being compared with those of the other students using percentile ranks. It indicates the percentage of scores that a given value is higher or greater than the others. To get the percentile rank of the value x in the given data, then: 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒗𝒂𝒍𝒖𝒆𝒔 𝒃𝒆𝒍𝒐𝒘 𝒙 + 𝟎. 𝟓 𝑷= ∗ 𝟏𝟎𝟎 𝒕𝒐𝒕𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒗𝒂𝒍𝒖𝒆𝒔 EXAMPLE #1: A 30-point quiz was given to 10 students and the scores are shown below. What is the percentile rank of 27? 23 25 19 21 28 15 20 24 22 27 SOLUTION: Arrange the data in ascending order. 15 19 20 21 22 23 24 25 27 28 There are eight values below 27. Then: 8+0.5 𝟖. 𝟓 𝑃= ∗ 100 = ∗ 𝟏𝟎𝟎 = 𝟎. 𝟖𝟓 ∗ 𝟏𝟎𝟎 = 85th PERCENTILE 10 𝟏𝟎 INTERPRETATION: This means that a student with a score of 27 performed better than 85% of the class. EXAMPLE #2: A 30-point quiz was given to 10 students and the scores are shown below. What is the percentile rank of 20? 23 25 19 21 28 15 20 24 22 27 SOLUTION: Arrange the data in ascending order. 15 19 20 21 22 23 24 25 27 28 There are two values below 20. Then: 𝟐+𝟎.𝟓 𝟐. 𝟓 𝑷= ∗ 𝟏𝟎𝟎 = ∗ 𝟏𝟎𝟎 = 𝟎. 𝟐𝟓 ∗ 𝟏𝟎𝟎 = 25th PERCENTILE 𝟏𝟎 𝟏𝟎 INTERPRETATION: This means that a student with a score of 20 performed better than 25% of the class. EXAMPLE #3: Find the percentile rank of a test score of 38 in the data set. 30 38 49 40 33 27 42 SOLUTION: Arrange the data in ascending order. 27 30 33 38 40 42 49 There are three values below 38. Then: 𝟑+𝟎.𝟓 𝟑. 𝟓 𝑷= ∗ 𝟏𝟎𝟎 = ∗ 𝟏𝟎𝟎 = 𝟎. 𝟓 ∗ 𝟏𝟎𝟎 = 50th PERCENTILE 𝟕 𝟕 INTERPRETATION: This means that a student with a score of 38 performed better than 50% of the class. EXAMPLE #4: Find the percentile rank of a test score of 40 in the data set. 30 38 49 40 33 27 42 25 SOLUTION: Arrange the data in ascending order. 25 27 30 33 38 40 42 49 There are five values below 40. Then: 𝟓+𝟎.𝟓 𝟓. 𝟓 𝑷= ∗ 𝟏𝟎𝟎 = ∗ 𝟏𝟎𝟎 = 𝟎. 𝟔𝟗 ∗ 𝟏𝟎𝟎 = 69th PERCENTILE 𝟖 𝟖 INTERPRETATION: This means that a student with a score of 40 performed better than 69% of the class. Quartiles Divides the data into four equal parts such as 1st quartile (𝑸𝟏 ), 2nd quartile (𝑸𝟐 ), and 3rd quartile (𝑸𝟑 ). EXAMPLE: 𝑸 𝟕+𝟖 𝟐= 𝟐 𝑸𝟐 = 𝟕. 𝟓 EXAMPLE #1: Find the 𝑄1 , 𝑄2 , and 𝑄3 of the following scores of students in a class. 11 14 25 30 27 18 13 28 17 26 SOLUTION: Arrange the data in ascending order. 11 13 14 17 18 25 26 27 28 30 Determine first the 𝑸𝟐 or the median. 𝑸 𝟏𝟖+𝟐𝟓 𝑸𝟐 = 𝟐𝟏. 𝟓 𝟐= 𝟐 For 𝑸𝟏 and 𝑸𝟑 11 13 14 17 18 25 26 27 28 30 𝑸𝟏 𝑸𝟐=𝟐𝟏.𝟓 𝑸𝟑 𝑸𝟏 = 𝟏𝟒 𝑸𝟐 = 𝟐𝟏. 𝟓 𝑸𝟑 = 𝟐𝟕 EXAMPLE #2: Find the 𝑄1 , 𝑄2 , and 𝑄3 of the following scores of students in a class. 10 17 9 7 11 13 5 6 3 12 14 2 4 SOLUTION: Arrange first in ascending order and determine the 𝑄2 or the median. 2 3 4 5 6 7 9 10 11 12 13 14 17 𝑸𝟐 For 𝐐𝟏 and 𝐐𝟑 2 3 4 5 6 7 9 10 11 12 13 14 17 𝟒+𝟓 𝑸𝟐 𝟏𝟐 + 𝟏𝟑 𝑸𝟏 = 𝑸𝟑 = 𝟐 𝟐 𝑸𝟏 = 𝟒. 𝟓 𝑸𝟑 = 𝟏𝟐. 𝟓 𝑸𝟏 = 𝟒. 𝟓 𝑸𝟐 = 𝟗 𝑸𝟑 = 𝟏𝟐. 𝟓 EXAMPLE #3: Find the 𝑄1 , 𝑄2 , and 𝑄3 of the following scores of students in a class. 10 17 9 7 11 13 5 6 3 12 14 2 SOLUTION: Arrange first in ascending order and determine the 𝑄2 or the median. 2 3 5 6 7 9 10 11 12 13 14 17 𝟗 + 𝟏𝟎 𝑸𝟐 = 𝑸𝟐 = 𝟗. 𝟓 𝟐 For 𝐐𝟏 , 𝐐𝟐 , and 𝐐𝟑 2 3 5 6 7 9 10 11 12 13 14 17 𝟓+𝟔 𝟏𝟐 + 𝟏𝟑 𝑸𝟏 = 𝑸𝟑 = 𝟐 𝟐 𝑸𝟏 = 𝟓. 𝟓 𝑸𝟐 = 𝟗. 𝟓 𝑸𝟑 = 𝟏𝟐. 𝟓 Z-score Also called as standard score. It is the number of standard deviations that a value is above or below the mean of the data set. Observed values above the mean have positive z- scores while values below the mean have negative z- scores. FORMULA FOR Z-SCORE where: EXAMPLE #1: June scored 83 in a quiz in Geometry for which the average score of the class is 78 with a standard deviation of 7. He also took a quiz in Calculus and scored 67 for which the average score of the class was 49, and the standard deviation was 11. Relative to other students in the class, did June do better in Geometry or Calculus?  SOLUTION: GEOMETRY CALCULUS 83 − 78 67 − 49 𝑧= 𝑧= 7 11 z = 0.714 z = 1.63 June scored 0.714 standard deviation above the mean in Geometry and 1.63 standard deviations above the mean in Calculus. Therefore, June performs better in Calculus. EXAMPLE #2: Listed below are the scores of Ruda in English, Science, Math, and Filipino with its mean and sd in every subject. SUBJECT SCORE MEAN sd English 35 29 4 Science 34 29 5 Math 30 22 7 Filipino 31 27 6 On what subject did Ruda performed best? a. English c. Math b. Science d. Filipino EXAMPLE #2: Listed below are the scores of Ruda in English, Science, Math, and Filipino with its mean and sd in every subject. SUBJECT SCORE MEAN sd English 35 29 4 Science 34 29 5 Math 30 22 7 Filipino 31 27 6 ENGLISH SCIENCE MATH FILIPINO 35 − 29 34 − 29 30 − 22 31 − 27 𝑧= 𝑧= 𝑧= 𝑧= 4 5 7 6 z = 1.5 z=1 z = 1.1 z = 0.7 Listed below are the scores of Ruda in English, Science, Math, and Filipino with its mean and sd in every subject. SUBJECT SCORE MEAN sd English 35 29 4 Science 34 29 5 Math 30 22 7 Filipino 31 27 6 On what subject did Ruda performed best? a. English Z-score = 1.5 c. Math Z-score = 1.1 b. Science Z-score = 1 d. Filipino Z-score = 0.7 Normal Distribution It shows a typical pattern that seems to be a part of many real-life phenomena. Normal distribution has a bell-shaped curve and is symmetric. It is symmetric around the mean: Two halves of the curve are the same in size. (mirror image) Normal Distribution Because of the exact symmetry of a normal curve, the center of a normal distribution is located at the highest point of the distribution and therefore, the mean, median, and mode are all equal. The total area under the curve is 1 or 100%. Normal Distribution LOWER 50% OF DATA UPPER 50% OF DATA Using the empirical rule of a normal distribution, approximately 68% of the data lie within 1 sd’s of the mean. 95% of the data lie within 2 sd’s of the mean. 99.7% of the data lie within 3sd’s of the mean. 0.15% 0.15% Normal Distribution Standard Normal Distribution Correlation It is a statistical method used to determine whether a linear relationship or association between variables exist. Scatter plot is used to describe the nature of the relationship between the variables. Scatter plot is a graph of the ordered pairs (x, y) of numbers consisting of the independent variable “x” along the x-axis and the dependent variable “y” along the y-axis. Correlation Positive Correlation Examples: Time spent of watching TV and Amount of Electric bill Price of gasoline and Amount of transport fare Negative Correlation Examples: Time spent of playing mobile games and Exam scores Speed and weight / Speed and time No Correlation (Zero Correlation) Examples: Shoe size and Intelligence Quotient Exam Scores and Gender