Part 3: Numerically Summarizing Data

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Part 3: Numerically Summarizing Data 3.1 Measures of Central Tendency 3.2 Measures of Dispersion 3.3 Measures of Central Tendency and Dispersion from Grouped Data 3.4 Measures of Position 3.5 The Five-Number Summary and Boxplots September 25, 2008

The Mean of a Set Section 3.1

Remark

The Median of a Set at the end of the day, the middle is the midpoint of the perceptions when they are requested from littlest to biggest or the other way around .

Example 1 Find the mean and middle of the arrangement of perceptions: {20, - 3, 4, 10, 6, - 1}.

Example 2 Find the mean and middle of the arrangement of perceptions: {-10, - 6 ,0, 4, 9}.

Mean and Dot Plot Notice that the mean is a support for the circulation of point masses on the lever (x-hub).

Add Points ("Weights") The support has moved 1.2 units to one side.

Shape, Mean and Median

Outlier An anomaly is a perception (information point) that falls well above or underneath the general arrangement of information. The mean can be very impact by an exception. The middle is said to be impervious to exceptions i.e ., it esteem is not changed essentially by the expansion or evacuation of an anomaly.

Example

Mode The mode is the most incessant perception of the variable. It is frequently utilized with unmitigated information. For numerical information, it can be utilized when the information is discrete. The method of the unmitigated variable shading is 35 ( red ).

Example Mia Hamm, who resigned at the 2004 Olympics, is thought to be the most productive player in worldwide soccer. He is a rundown of the quantity of objectives scored over her 18-year profession. MHG = {0,0,0,4,10,1,10,10,19,9,18,20,13,13,2,7,8,13}. Considering the populace as the quantity of objectives scored by Mia Hamm, locate the mean and middle and method of this set.

Mean, Median and Mode and Distribution Shape

Measures of Dispersion Consider the accompanying arrangements of perceptions: S 1 = {0,0,0,0,0,0,0,0,0,0} S 2 = {-5,- 4,- 3,- 2,- 1,1,2,3,4,5}. Both sets have a similar mean and middle (specifically, 0). In any case, the histograms or speck plots are very unique. However, their speck plot is altogether different. See that the distinction between the littlest and biggest number in each set is very extraordinary. Segment 3.2

Range of a Set of Observations Remark: The range is totally dictated by just two purposes of the arrangement of perceptions.

Example Lance Armstrong won the Tour de France seven sequential circumstances (1999-2005). Here is information about his triumphs. The reaches for every classification of winning will be: Winning Time: run = 92.552 - 82.087 = 10.465 Distance: extend = 3687 - 3278 = 409 Winning Speed: go = 41.65 - 39.46 = 2.19 Winning Margin: run = 7.283 - 1.017 = 6.266

The Spread of Quantitative Data Consider the recurrence circulations of two distinct informational collections. See how the tails of every conveyance change from being near one another to being far separated. Segment 2.4

The Deviation from the Mean

Variance and Standard Deviation Definition: The "normal" of the square of all deviations in a specimen is known as the change of the example. The standard deviation of a specimen is characterized as the square foundation of the change. Address: Why n - 1 rather than n in these recipes?

Remark There is a heartbreaking deception on how the words, change and standard deviation, are utilized. These amounts are figured diverse routes, contingent upon whether the set under thought is a populace or a specimen of a populace. Things being what they are whether we utilize the recipes for change and standard deviation where we separate by n rather than n-1 , then the standard deviation of the example will reliably belittle the standard deviation of the populace. This is called inclination. Subsequently, we will some of the time utilize the accompanying definitions and will recognize test standard deviation and populace standard deviation.

Example For the arrangement of perceptions (test), {0,- 3,10,7,5,- 3,0}, Find the scope of the specimen. Locate the mean and middle of the specimen. Discover the change of the specimen. Locate the standard deviation of the specimen.

Example For the two arrangement of perceptions, S = {-1,0,0,0,1} and T = {-1,- 1,- 1,- 1,0,1,1,1,1}, Find the mean and middle for each set. Locate the standard deviation for each set. We see from the spot plot that the set T has more focuses that fluctuate from the mean and thus, has a bigger standard deviation.

Properties of the Standard Deviation The bigger the spread (variety) in the information, the bigger the standard deviation. The standard deviation is zero just if and just if the set from which it is registered has the greater part of its components the same in which case the mean of the set is this number. The standard deviation is affected by exceptions. This is genuine in light of the fact that the deviation from the mean of the set to the anomaly is an expansive number in supreme esteem. The standard deviation yields more data than the scope of the set. (Why?)

Example The accompanying information speaks to the strolling time (in minutes) from the quarters or loft to Professor Bisch's course on administrator algebras. We regard the nine understudies as the number of inhabitants in Prof. Bisch's class. Discover the populace mean and standard deviation. Pick an example of 4 and register the mean and standard deviation of the specimen.

Bell-molded (symmetric) Distributions Consider an arrangement of perceptions that is ringer formed. Every one of the three circulations have diverse standard deviations.

Empirical Rule for nearly Bell-molded Distributions

Caution The Empirical Rule for ringer formed conveyances is an observational law, not a reality. The better the circulation is by and large impeccably ringer molded, then better the precision of the law. It is valuable in disclosing to us how the information is thought about the mean of the circulation.

Example

Detailed Empirical Rule

Example The dissemination of the length of jolts delivered by the Acme Bolt Company is roughly ringer formed with a mean of 4 inches and a standard deviation of 0.007 inches. What is the scope of length for 68% of the jolts delivered by this organization? What rate of jolts will be between 3.986 inches and 4.014 inches? On the off chance that the organization disposes of any jolts that are under 3.986 inches or more noteworthy than 4.014 inches, what rate of jolts will be disposed of? What rate of the jolts will be between 4.007 inches and 4.021 inches?

Chebyshev Inequality Example: Suppose that a populace has a mean of 73.5 and a standard deviation of 5.5. Discover an interim that contains no less than 75% of the information focuses in the populace.

Example In December 2004, the normal cost of customary unleaded gas barring charges in the United States as $1.37 per gallon. Specialists in the Department of Energy evaluated that the standard deviation at this mean cost was $0.05. Utilizing Chebyshev's Inequality,estimate the rate of gas stations that had costs inside 3 standard deviations of the mean? What rate had costs inside 2.5 standard deviations?

Remark Chebyshev's Inequality does not put any preconditions on the state of the informational collection. It is valid for populaces and tests. The hypothesis does not state that there are precisely 100(1-1/k 2 )% focuses in an interim that is one standard deviation from the mean, yet rather there are at any rate this number.

Mean and Standard Deviation for Grouped Data Section 3.3

Example

Example

Weighted Mean of a Set Given an arrangement of numbers, assume that we trust that a portion of the numbers are more essential than different numbers in the set. To mirror this documentation, we characterized the weighted mean of an arrangement of numbers.

Example Consider the set S = {-3, 1, 0, 3, - 1, 1, 0} and the weights {1.5, 0, 1, - 1, 1, 2, 1}. Locate the weighted mean of this set concerning the given weights.

Approximation for Standard Deviation and Variance for Grouped Data

Example

Approximating the Median of gathered Data

Example

Measures of Position in a Distribution The mean and middle give us data about the "inside" of an arrangement of perceptions (the dispersion). The range and standard deviation give us data about the "spread" of the conveyance. We now present an idea that is proportional to the "position" in a dispersion. It will utilize the idea of percentiles . The percentile will how the dispersion can be isolated into parts (once in a while rise to) which thusly will give us the thought of position inside the dissemination. Area 3.4

z-score

Example: Consider the specimen: {-1,0,1,5,19}. Process the z - score for every information point.

Application of z-score The normal 20-to 29-year old man is 69.6 inches tall with a standard deviation of 2.7 inches. The normal 20-to 29-year old lady is 64.1 crawls with a standard deviation of 2.6 inches. As for their populace, who is generally taller: a 75-inch man or a 70-inch lady?

Percentile Definition: The k th percentile in an appropriation, P k , is a number that is the rate of the perceptions that fall beneath or at this esteem. As it were, it subdivides the aggregate region encased by the circulation into two sub-territories, A 1 and A 2 , so that aggregate region is partitioned into two sections: k and 100-k .

Algorithm for Percentiles

Example Find the 20 th percentile of the set: S = {-1,0,3,5,9,12,15,18,25}. Next discover the 45 th percentile.

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