Information about CABT Math 8 measures of central tendency and dispersion

A Brief Introduction to Statistics Branches of Statistics Two branches of statistics: 1. Descriptive Statistics: Describes the characteristics of a product or process using information collected on it. 2. Inferential Statistics (Inductive): Draws conclusions on unknown process parameters based on information contained in a sample. Uses probability

A Brief Introduction to Statistics Data DATA is any quantitative or qualitative information. Types of Data: 1. Quantitative – numerical information obtained from counting or measuring (e.g. age, qtr. exam scores, height) 2. Qualitative – descriptive attributes that cannot be subjected to mathematical operations (e.g. gender, religion, citizenship)

Measures of Central Tendency and Dispersion The Measures of Central Tendency and Dispersion Statistics use numerical values used to summarize and compare sets of data. Measure of Central Tendency: number used to represent the center or middle set of a set of data Measure of Dispersion (or Variability): refers to the spread of values about the mean. (i.e., how spread out the values are with respect to the mean)

The Measures of Central Tendency

Measures of Central Tendency and Dispersion Measures of Central Tendency The Measure of Central Tendency: 1. Mean - the (arithmetic) average (or the sum of the quantities divided by the number of quantities) Median – the middle value of a set of ordered data 3. Mode – number in a data set that occurs most frequently 2.

Measures of Central Tendency and Dispersion The Mean It‘s known as the typical ―average.‖ It is the most common measure of central tendency. Symbolized as: ◦ x for the mean of a sample ◦ μ (Greek letter mu) for the mean of a population • It‘s equal to the sum of the quantities in the data set divided by the number of quantities x x n

Measures of Central Tendency and Dispersion The Mean Example 1 Find the mean of the numbers in the following data sets: a. b. 3, 5, 10, 4, 3 x 3 5 10 4 3 5 85, 87, 89, 90, 91, 98 x 540 6 90 25 5 5

Measures of Central Tendency and Dispersion The Mean Example 2 The table on the right shows the age of 13 applicants for a job in a factory in EPZA. What is the average age of the applicants? (Adapted from DOLE-BLES i-Learnstat module on Measures of Central Tendency) Solution: x 318 13 24.5

Measures of Central Tendency and Dispersion The Weighted Mean It is a mean where some values contribute more than others. Each quantity is assigned a corresponding WEIGHT (e.g. frequency or number, units, per cent) The weighted mean is equal to the sum of the products of the quantities (x) and their corresponding weights (w), divided by the sum of the weights. x wx w

Measures of Central Tendency and Dispersion The Weighted Mean Example 3 SCORE NO. OF STUDENTS 5 8 4 6 3 3 2 2 1 1 The table shows the scores of 20 students in a 5-item Math IV seatwork. Find the average score of the class.

Measures of Central Tendency and Dispersion The Weighted Mean Example 3 SolutionSCORE x 78 20 3.9 PRODUCT 5 8 40 4 6 24 3 3 9 2 2 4 1 Multiply the scores by the number of students, then find the sum. Finally, divide by the total number of students The average score is NO. OF STUDENTS 1 1 sums 20 78

Measures of Central Tendency and Dispersion The Median Used to find the middle value (center) of a distribution. Used when one must determine whether the data values fall into either the upper 50% or lower 50% of a distribution. Used when one needs to report the typical value of a data set, ignoring the outliers (few extreme values in a data set). ◦ Example: median salary, median home prices in a market

Measures of Central Tendency and Dispersion The Median How to find the median: Order the data in increasing order. If the number of data is ODD, the median is the middle number. If n is odd, the middle number in n observations is the (n + 1)/2 th observation If the number of data is EVEN, the median is the mean of the two middle numbers. If n is even the middle number in n observations is the average of the (n/2)th and the (n/2+1)th observation

Measures of Central Tendency and Dispersion The Median Example 4 Find the median of each set of data. a. 1, 2, 2, 3, 3, 4, 4, 5, 5 b. 1, 2, 2, 3, 3, 4, 4, 4, 5, 5 Answers a. Me = 3 (the 5th number) b. The average of 5th and 6th numbers: 3 4 Me 2 3.5

Measures of Central Tendency and Dispersion The Median Example 5 Find the median of the following: 3, 5, 6, 10, 9, 8, 7, 8, 9, 10, 7, 2, 5, 7 Solution: Arrange from lowest to highest: 2, 3, 5, 5, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10 The median is 7.

Measures of Central Tendency and Dispersion The Mode It is the number that appears most frequently in a set of data. It is used when the most typical (common) value is desired. It is not always unique. A distribution can have no mode, one mode, or more than one mode. When there are two or more modes, we say the distribution is multimodal. (for two modes, we say that the distribution is bimodal)

Measures of Central Tendency and Dispersion The Mode Example 6 The table shows the scores of 20 students in a 5-item AP quiz. SCORE NO. OF STUDENTS 5 6 4 7 3 4 What is the modal score? 2 2 Answer: 4 1 1

Measures of Central Tendency and Dispersion The Mode Example 7 Find the mode of each set of data. a. 1, 2, 2, 3, 3, 4, 4, 4, 5, 5 Mo = 4 a. 1, 2, 2, 3, 3, 3,4, 4, 4, 5, 5Mo = 3 and 4 a. 1, 2, 3, 4, 5 No mode

Measures of Central Tendency and Dispersion The Mode Example 8 Find the mode of the following: 3, 5, 6, 10, 9, 8, 7, 8, 9, 10, 7, 2, 5, 7 Solution: Arrange from lowest to highest: 2, 3, 5, 5, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10 The mode is 7.

Measures of Central Tendency and Dispersion Check your understanding 4, 8, 12, 15, 3, 2, 6, 9, 8, 7 The data set above gives the waiting times (in minutes) of 10 students waiting for a bus. Find the mean, median, and mode of the data set.

Measures of Central Tendency and Dispersion Check your understanding 4, 8, 12, 15, 3, 2, 6, 9, 8, 7 The data set above gives the waiting times (in minutes) of 10 students waiting for a bus. Find the mean, median, and mode of the data set. Solution Arrange the data first in increasing order: 2, 3, 4, 6, 7, 8, 8, 9, 12, 15 Mean : x 74 10 7.4 min Median : Me Mode : Mo 8 min 7 8 2 7.5 min

The Measures of Dispersion

Measures of Central Tendency and Dispersion Measures of Dispersion The Measure of Dispersion or Variability 1. Range – the difference of the largest and smallest value 2. Mean Absolute Deviation – the average of the positive differences from the mean 3. Standard deviation – involves the average of the squared differences from the mean. (related: variance)

Measures of Central Tendency and Dispersion Range Simply the difference between the largest and smallest values in a set of data Useful for analysis of fluctuations and for ordinal data Is considered primitive as it considers only the extreme values which may not be useful indicators of the bulk of the population. The formula is: Range = largest observation - smallest observation

Measures of Central Tendency and Dispersion Range Example 10 Find the range of the following data sets: a. 3, 5, 10, 4, 3 range 10 3 7 b. 85, 87, 89, 90, 91, 98 range 98 85 13 c. 3, 5, 6, 10, 9, 8, 7, 8, 9, 10, 7, 2, 5, 7 range 10 2 8

Measures of Central Tendency and Dispersion Mean Deviation It measures the ‗average‘ distance of each observation away from the mean of the data Gives an equal weight to each observation Generally more sensitive than the range, since a change in any value will affect it The formula is MD x x n x where x is a quantity in the set, and n is the number of data. is the mean,

Measures of Central Tendency and Dispersion Mean Deviation To find the mean deviation:MD 1. 2. x x n Compute the mean. Get all the POSITIVE difference of each number and the mean. (It‘s the same as getting the absolute value of each difference) 3. 4. Add all the results in step 2. Divide by the number of data.

Measures of Central Tendency and Dispersion Mean Deviation Example 11 Find the mean deviation of 3, 6, 6, 7, 8, 11, 15, 16 Solution x STEP 1: Find the mean: 72 8 9

Measures of Central Tendency and Dispersion Mean Deviation VALUE Example 11 Find the mean deviation of 3, 6, 6, 7, 8, 11, 15, 16 STEP 2: Find the POSITIVE difference of each number and the mean (9). POSITIVE DIFFERENCE 3 6 6 3 6 7 8 11 15 16 3 2 1 2 6 7

Measures of Central Tendency and Dispersion Mean Deviation VALUE Example 11 Find the mean deviation of 3, 6, 6, 7, 8, 11, 15, 16 STEP 3: Add all the differences. POSITIVE DIFFERENCE 3 6 6 3 6 7 8 11 15 16 3 2 1 2 6 7 sum 30

Measures of Central Tendency and Dispersion Mean Deviation VALUE Example 11 Find the mean deviation of 3, 6, 6, 7, 8, 11, 15, 16 STEP 4: Divide the result by the number of data to get the MD: MD 30 8 3.75 POSITIVE DIFFERENCE 3 6 6 3 6 7 8 11 15 16 3 2 1 2 6 7 sum 30

Measures of Central Tendency and Dispersion Mean Deviation What does the answer in the previous example mean? It means that the quantities have an average difference of 3.75 from the mean (plus or minus).

Measures of Central Tendency and Dispersion Standard Deviation Measures the variation of observations from the mean The most common measure of dispersion Takes into account every observation Measures the ‗average deviation‘ of observations from the mean Works with squares of residuals, not absolute values—easier to use in further

Measures of Central Tendency and Dispersion Standard Deviation The formula for the standard deviation is 2 x x n where x is a quantity in the set, x is the mean, and n is the number of data.

Measures of Central Tendency and Dispersion Variance The variance is simply the square of the standard deviation, or 2 Variance : 2 x x n 2

Measures of Central Tendency and Dispersion Standard Deviation x x To find the standard deviation: n 1. Compute the mean. 2. Get the difference of each number and the mean. 3. Square each difference 4. Add all the results in step 3. 5. Divide by the number of data. 6. Get the square root. Note: If the VARIANCE is to be computed, skip the last step. 2

Measures of Central Tendency and Dispersion Standard Deviation Population versus Sample Standard Deviation The standard deviation used here is called the POPULATION standard deviation. For very large populations, the SAMPLE standard deviation (s) is used. Its 2 formula is x x s n 1

Measures of Central Tendency and Dispersion Standard Deviation Alternative Formula for the Standard Deviation formula for standard deviation Another uses only the sum of the data as well the sum of the squares of the data. This is n x 2 x n 2

Measures of Central Tendency and Dispersion Standard Deviation To find the standard deviation using the alternative formula: n x x n 1. Compute the squares of the data. 2. Get the sum of the data and the sum of the squares of the data. 3. Multiply the sum of the squares by the number of data, then subtract to the square of the sum of the data. 4. Get the square root of the result in step 3. 5. Divide the result by the number of data. 2 2

Measures of Central Tendency and Dispersion Standard Deviation Example 12 Find the standard deviation of 3, 6, 6, 7, 8, 11, 15, 16 using the given and the alternative formulas. Solution Before using the formulas, it‘s better to tabulate all results.

Measures of Central Tendency and Dispersion Standard Deviation Using the given formula x x–x x x 2 n (x – x)2 3 –6 36 6 6 7 8 11 15 16 –3 –3 –2 –1 2 6 7 sum 9 9 4 1 4 36 49 148 x x n 148 8 4.3 2

Measures of Central Tendency and Dispersion Standard Deviation Using the alternative formula x 3 sum x2 9 6 6 7 8 11 15 16 72 36 36 49 64 121 225 256 796 n x2 x 2 n n x 2 x 2 n 8 796 72 2 8 1 ,184 8 4.3 Ano ang pipiliin mo?

Measures of Central Tendency and Dispersion Standard Deviation Remark: For both cases, the variance is simply the square of the standard deviation. The value2is 74 Woohoo…

Measures of Central Tendency and Dispersion Check your understanding Find the standard deviation and variance of the following data set: 4, 8, 12, 15, 3, 2, 6, 9, 8, 7

Measures of Central Tendency and Dispersion

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