Measures of Central Tendency PDF

**What is Measures of Central Tendency?** A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendencies are sometimes called measures of central location. They are also cla...

**What is Measures of Central Tendency?** A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendencies are sometimes called measures of central location. They are also classed as summary statistics. **Module Outcome/s** After this learning module, students will be able to: 1\. Learn the properties of mean, median and mode 2\. Analyze and interpret the data presented in the table using measures of central tendency 3\. Articulate the importance of mathematics in one's life. 4\. Express appreciation for mathematics as a human endeavor. 5\. Express appreciation for Mathematics as a human endeavor. **What is Median?** A median can define as the middle value/observation in an organized list of numbers and falls in the middle-most position of the whole data. Properties of Median: 1. The median is unique, there is only one median for a set of data. 2. The median can be determined by arranging the set of data from lowest to highest (or highest or lowest). 3. Median is not affected by the extreme small or large values. 4. Median can be applied for ordinal, interval, and ratio data. 5. Median is most appropriate in a skewed data. **UNGROUPED DATA** The median value in an *ungrouped data* is determined by first arranging the numbers in value order from lowest to highest or vice versa. If there is an odd amount of numbers, the median value is the middle most number, with the same amount of numbers below and above. If there is an even amount of numbers in the list, the middle pair must be determined, added together and divided by two to find the median value. The median can be used to determine an approximate average. To determine the value of median for ungrouped, we need to consider two rules: 1\. If n is odd, the median is the middle ranked. 2\. If n is even, then the median is the average of the two-middle ranked values. **Median (Rank Value) =** [\$\\frac{\\mathbf{n + 1}}{\\mathbf{2}}\$]{.math.inline} *Note that n is the population/sample size.* **Example 1: Find the median of the ages of eleven middle-call center agents of Sutherland. The ages are 30, 38, 28, 35, 25, 24, 33, 26, 32, 31 and 37.** **Solution:** Step 1: Arrange the data in order. 24, 25, 26, 28, 30, 31, 32, 33, 35, 37, 38. Step 2: Select the middle rank value. Median (Rank Value) = [\$\\frac{n + 1}{2} = \\frac{11 + 1}{2} = \\frac{12}{2} = 6\$]{.math.inline} Step 3: Identify the median in the data set, 24, 25, 26, 28, 30, 31, 32, 33, 35, 37, 38 or [24, 25, 26, 28, 30,] 31, [32, 33, 35, 37, 38] Hence, the median age is 31 years. **Example 2: The daily rates of the ten Sutherland call-center agents are ₱850, ₱920, ₱700, ₱830, ₱670, ₱740, ₱870, ₱650, ₱590, ₱770. Find the median daily rate of employee.** **Solution:** Step 1: Arrange the data in order. ₱590, ₱650, ₱670, ₱700, ₱740, ₱760, ₱830, ₱850, ₱870, ₱920 Step 2: Select the middle rank value. Median (Rank Value) = [\$\\frac{n + 1}{2} = \\frac{10 + 1}{2} = \\frac{11}{2} = 5.5\$]{.math.inline} ₱590, ₱650, ₱670, ₱700, ₱740, ₱760, ₱830, ₱850, ₱870, ₱920 **Or** 740 +760 = 1500 1500 ÷ 2 = **750** Since the middle point falls between ₱740 and ₱760, we can determine the median of the data set by getting the average of the two values. Therefore, the median daily rate is ₱750. **What is Mode?** The number/value/observation in a data set which appears the greatest number of times. If no number in the list is repeated, then there is no mode for the list. However, it is also possible to have more than one mode for the same distribution of data, (bi-modal, tri-modal, or multi-modal). KINDS OF MODE A data set that has only one value that occurs the greatest frequency is called as **unimodal**. If the data has two values with the same greatest frequency, both values are considered the mode, and the data set is **bimodal**.  If the data set has more than two modes, then the data set is said to be **multimodal.** Properties of Mode: 1\. The mode can be found by locating the most frequently occurring value. 2\. The mode is the easiest average to compute. 3\. There can be more than one mode or even no mode in any given data set. 4\. Mode is not affected by extreme values 5\. Mode can be applied for nominal, ordinal, interval, and ratio data **UNGROUPED DATA** To find the mode of an ungrouped data, find the frequency of each number/value/observation in the given data set. Then, choose the number/value/observation having the highest frequency as the mode. Mode = number/value/observation with the highest frequency **Example 1: The following data represent the age of the BSE 1Q students, 19, 18, 20, 20, 21, 17, 20, 25. Find the mode.** Solution: Step 1: Arrange the data set to ascending or descending order. 17, 18, 19, 20, 20, 20, 21, 25 Step 2: Determine the number that appeared the greatest number of times. 17, 18, 19, [20, 20, 20], 21, 25 So, looking at the given data above the number that appeared the greatest number of times is the number 20. **Example 2: Mrs. Romero conduct a test for her Students in BSE 1Q in mathematics subject. The following data are the scores that her students got from the test 20, 18, 19, 25, 20, 21, 20, 25, 30, 29, 28, 29, 25, 25, 27, 26, 22 and 20. Find the mode of the given data set.** Solution: Step 1: The ordered array for these data is 18, 19, 20, 20, 20, 20, 21, 22, 25, 25, 25, 25, 26, 27, 28, 29, 29, 30. Step 2: Determine the number that appeared the greatest number of times. 18, 19, [20, 20, 20, 20], 21, 22[, 25, 25, 25, 25], 26, 27, 28, 29, 29, 30 So, looking at the given data above, there are two modes 20 and 25 since each of these values occurs four times.