Mathematics in the Modern World GNED 03 PDF

SLIDESMANIA.COM MATHEMATICS IN THE MODERN WORLD GNED 03 SLIDESMANIA.COM Measures of Central Tendency One of the most basic statistical concepts involves finding...

SLIDESMANIA.COM MATHEMATICS IN THE MODERN WORLD GNED 03 SLIDESMANIA.COM Measures of Central Tendency One of the most basic statistical concepts involves finding measures of central tendency of a set of numerical data. It is often helpful to find numerical values that locate, in some sense, the center of a set of data. We will consider three types of averages, known as the arithmetic mean, the median, and the mode. Each of these averages is a measure of central tendency for the numerical data. SLIDESMANIA.COM Measures of Central Tendency The Arithmetic Mean It is the most commonly used measure of central tendency. The arithmetic mean of a set of numbers is often referred to as simply the mean. To find the mean for a set of data, find the sum of the data values and divide by the number of data values. σ 𝑥𝑖 𝑀𝑒𝑎𝑛 = 𝑛 SLIDESMANIA.COM Measures of Central Tendency The Arithmetic Mean (Example) Six friends in a biology class of 20 students received test grades of 92, 84, 65, 76, 88, and 90. Find the mean of these test scores. SLIDESMANIA.COM Measures of Central Tendency The Arithmetic Mean (Example) A doctor ordered 4 separate blood tests to measure a patient’s total blood cholesterol levels. The test results were 245, 235, 220, and 210. Find the mean of the blood cholesterol levels. SLIDESMANIA.COM Measures of Central Tendency The Median Essentially, it is the middle number or the mean of the two middle numbers in a list of numbers that have been arranged in numerical order from smallest to largest or largest to smallest. SLIDESMANIA.COM Measures of Central Tendency The Median (Example) Find the median of the following lists: a. 4, 8, 1, 14, 9, 21, 12 b. 46, 23, 92, 89, 77, 108 SLIDESMANIA.COM Measures of Central Tendency The Mode The mode of a list of numbers is the number that occurs most frequently. Some lists of numbers do not have a mode. For instance, in the list 1, 6, 8, 10, 32, 15, 49 , each number occurs exactly once. Because no number occurs more often than the other numbers, there is no mode. A list of numerical data can have more than one mode. For instance, in the list 4, 2, 6, 2, 7, 9, 2, 4, 9, 8, 9, 7, the number 2 and the number 9 both occur three times. Thus, 2 and 9 are both modes for the data. SLIDESMANIA.COM Measures of Central Tendency The Mode (Example) Find the mode of the data in the following lists: a. 18, 15, 21, 16, 15, 14, 15, 21 b. 2, 5, 8, 9, 11, 4, 7, 23 SLIDESMANIA.COM Measures of Central Tendency The mean, the median, and the mode are all averages; however, they are generally not equal. The mean of a set of data is the most sensitive of the averages. A change in any of the numbers changes the mean, and the mean can be changed drastically by changing an extreme value. In contrast, the median and the mode of a set of data are usually not changed by changing an extreme value. When a data set has one or more extreme values that are very different from the majority of data values, the mean will not necessarily be a good indicator of an average value. SLIDESMANIA.COM Measures of Central Tendency The Weighted Mean A value called the weighted mean is often used when some data values are more important than others. Consider the situation in which a professor counts the final exam score as 2 test scores. To find the weighted mean of the student’s scores, the professor first assigns a weight to each score. In this case, the professor could assign each of the test scores a weight of 1 and the final exam score a weight of 2. SLIDESMANIA.COM Measures of Central Tendency The Weighted Mean A student with test scores of 65, 70, and 75 and a final examination score of 90 has a weighted mean of 65 × 1 + 70 × 1 + 75 × 1 + (90 × 2) 390 = = 78 5 5 SLIDESMANIA.COM Measures of Central Tendency The Weighted Mean The weighted mean of 𝑛 numbers 𝑥1 , 𝑥2 , 𝑥3 , … , 𝑥𝑛 with respective assigned weights 𝑤1 , 𝑤2 , 𝑤3 , … , 𝑤𝑛 is σ(𝑥𝑖 𝑤𝑖 ) 𝑊𝑒𝑖𝑔ℎ𝑡𝑒𝑑 𝑚𝑒𝑎𝑛 = σ 𝑤𝑖 SLIDESMANIA.COM Measures of Central Tendency The Weighted Mean (Example) Consider the grade of a freshman student during the first semester. Subject Units Grade Purposive Communication 3 2.25 STS 3 1.75 MMW 3 2.00 Panitikan 3 2.00 PHED 1 2 1.50 Military Science 1.5 1.25 SLIDESMANIA.COM Measures of Dispersion – quantities that describe the spread or variability of the observations in a given data set Some common measures: Range Variance Standard Deviation Coefficient of Variation SLIDESMANIA.COM Measures of Dispersion Range The difference between the maximum and minimum values in a data set. 𝑅 = 𝑀𝐴𝑋 − 𝑀𝐼𝑁 Some Properties of the Range The larger the value of the range, the more dispersed the observations are. It is quick and easy to understand. SLIDESMANIA.COM Measures of Dispersion Range (Example) Heartbeats of 15 male GNED 03 students: 54, 58, 58, 60, 62, 65, 66, 71, 74, 75, 77, 78, 80, 82, 85 SLIDESMANIA.COM Measures of Dispersion Variance The average squared difference of the observations from the mean. σ 𝑁 2 2 𝑖=1 𝑥𝑖 − 𝜇 𝜎 = 𝑁 𝑛 2 2 σ 𝑖=1 𝑥𝑖 − 𝑥ҧ 𝑠 = 𝑛−1 SLIDESMANIA.COM Measures of Dispersion Some Properties of the Variance One of the most useful measures of dispersion. All observations contribute in the computation. Always nonnegative. Becomes larger as the observations become more dispersed from the mean. Comes in the square of the unit of measure of the given set of values. SLIDESMANIA.COM Measures of Dispersion Variance (Example) Heartbeats of 15 male GNED 03 students: 54, 58, 58, 60, 62, 65, 66, 71, 74, 75, 77, 78, 80, 82, 85 SLIDESMANIA.COM Measures of Dispersion Standard Deviation – The average deviation of the observations from the mean. – The positive square root of the variance. – The unit of measure is the same as that of the observations. σ𝑁 𝑖=1 𝑥𝑖 − 𝜇 2 𝜎= 𝑁 σ𝑛𝑖=1 𝑥𝑖 − 𝑥ҧ 2 𝑠= 𝑛−1 SLIDESMANIA.COM Measures of Dispersion Standard Deviation (Example) Heartbeats of 15 male GNED 03 students: 54, 58, 58, 60, 62, 65, 66, 71, 74, 75, 77, 78, 80, 82, 85 SLIDESMANIA.COM Measures of Dispersion Coefficient of Variation A relative measure that indicates the magnitude of variation relative to the magnitude of the mean expressed in percent. 𝜎 𝐶𝑉 = × 100 𝜇 𝑠 𝐶𝑉 = × 100 𝑥ҧ SLIDESMANIA.COM Measures of Dispersion Some Properties of the Coefficient of Variation It is unitless. It is used to compare dispersion of two or more data sets with the same or different units. The higher the 𝐶𝑉, the more variable is the data set relative to its mean. SLIDESMANIA.COM Measures of Dispersion Coefficient of Variation (Example) Heartbeats of 15 male GNED 03 students: 54, 58, 58, 60, 62, 65, 66, 71, 74, 75, 77, 78, 80, 82, 85  Thank you! SLIDESMANIA.COM