# Section 2

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﻿Section 2 Describing circulations with numbers

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Chapter Outline 1. Measuring focus: the mean 2. Measuring focus: the middle 3. Looking at the mean and the middle 4. Measuring spread: the quartiles 5. The five-number synopsis and boxplots 6. Measuring spread: the standard deviation 7. Picking measures of focus and spread

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Measuring focus: the mean Notation: It is essentially the standard number-crunching normal. Assume that we have n perceptions (information measure, number of people). Perceptions are indicated as x 1 , x 2 , x 3 , … x n .

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Measuring focus: the mean How to get ? Case 2.1 (P.33)

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Measuring focus: the middle Notation: M Median M is the midpoint of a dissemination ��  a large portion of the perceptions are littler than M and the other half are bigger than M.

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Measuring focus: the middle How to discover M? 1. Sort all perceptions in expanding request (This progression is important!!!) 2. In the event that n is odd, perception is M. on the off chance that n is even, normal of two focus qualities is M. Take note of that is the area of the middle in the requested rundown, not the middle esteem.

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Measuring focus: the middle Examples Case 1. 11, 21, 13, 24, 15, 26, 17 Case 2. 11, 21, 13, 24, 15, 26 Example 2.2, 2.3 (P.35)

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Mean versus Middle Median is more safe than the mean. The mean and middle of a symmetric conveyance are near one another. On the off chance that the appropriation is precisely symmetric, the mean and middle are precisely the same. In a skewed conveyance, the mean is more distant in the long tail than is the middle. Case 1, 2, 3, 4, 5, 6, 10000

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Inference : Strongly skewed disseminations are accounted for with middle than the mean.

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Measuring Spread: The Quartiles The quartiles stamp out the center portion of the dissemination.

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Calculating the Quartiles : Step1 . Mastermind the perceptions in expanding request and find the middle M in the requested rundown of perceptions. Step2 . The main quartile Q1 is the middle of the perceptions whose position in the requested rundown is to one side of the area of the general middle. S tep3 . The third quartile Q3 is the middle of the perceptions whose position in the requested rundown is to one side of the area of the general middle.

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Measuring spread: the quartiles Example 2.4 (P. 37) Example 2.5 (P. 38) Note: (1) It is essential to sort information first before we attempt to discover quartiles! (2) Quartiles are safe.

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The five-number synopsis and boxplots The five-number outline: Minimum, Q 1 , M, Q 3 , Maximum. Boxplot is a chart of five number synopsis. Boxplots are most valuable for one next to the other examination of a few dispersions.

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Boxplot 1. A boxplot is a chart of the five-number outline 2. A focal box traverses the quartiles 3. A line in the crate denote the middle 4. Lines reached out from the container out to the base and most extreme 5. Extend = most extreme - least

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The five-number outline and boxplot Figure 2.2(P.39): one next to the other boxplots looking at the dispersions of acquiring for two levels of instruction.

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The five-number synopsis and boxplots

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Inference : Boxplot likewise gives a sign of the symmetry or skewness of a dissemination . - In a symmetric conveyance Q1 and Q3 are similarly inaccessible from the middle, yet if there should be an occurrence of right skewed one the third quartile would be further over the middle than the main quartile cry it.

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Measuring spread: the standard deviation It says how far the perceptions are from their mean. The change s 2 of an arrangement of perceptions is a normal of the squares of the deviations of the perceptions from their mean. Documentation: s 2 for change and s for standard deviation

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Why (n-1) ? As the entirety of the deviations dependably breaks even with 0, so the learning of (n-1) of them decides the last one. - Only (n-1) of the squared deviations are variable however not the last one, so we normal by separating the aggregate by (n-1). The number (n-1) is known as the degrees of opportunity of the fluctuation or standard deviation

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Measuring spread: the standard deviation To discover the change and the standard deviation 1. Locate the mean of the information set 2. Subtract the mean from every number (we call that deviation) 3. Square every outcome 4. Whole all the square 5. Isolate the entirety of square by n-1, where n is the quantity of all perceptions. Presently you get change 6. Standard deviation is only the positive square base of the difference.

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Measuring spread: the standard deviation Example 2.6 (P.42)

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Properties of s 2 and s measures spread about the mean and ought to be utilized just when the mean is picked as the measure of focus. s 0 and s=0 just when each of the perception values does not vary from each other. S is not safe.

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Choosing measures of focus and spread With a skewed circulation or with a dispersion with outrageous exceptions, five-number outline is better. With a symmetric conveyance (without anomalies), mean and standard deviation are better.