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Part 1 THE MEAN VALUES

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СHAPTER QUESTIONS Measures of location Types of means Measures of location for ungrouped data - Arithmetic mean - Harmonic mean - Geometric mean - Median and Mode 4. Measures of location for grouped data - Arithmetic mean - Harmonic mean - Geometric mean - Median and Mode

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What is the mean? The mean - is a general indicator characterizing the typical level of varying trait per unit of qualitatively homogeneous population.

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Statistics derive the formula of the means of the formula of mean exponential: Statistics derive the formula of the means of the formula of mean exponential: We introduce the following definitions - X-bar - the symbol of the mean Х1, Х2...Хn – measurement of a data value f- frequency of a data values; n – population size or sample size.

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There are the following types of mean: There are the following types of mean: If z = -1 - the harmonic mean, z = 0 - the geometric mean, z = +1 - arithmetic mean, z = +2 - mean square, z = +3 - mean cubic, etc.

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The higher the degree of z, the greater the value of the mean. If the characteristic values are equal, the mean is equal to this constant. The higher the degree of z, the greater the value of the mean. If the characteristic values are equal, the mean is equal to this constant. There is the following relation, called the rule the majorizing mean:

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There are two ways of calculating mean: There are two ways of calculating mean: for ungrouped data - is calculated as a simple mean for grouped data - is calculated weighted mean

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Types of means

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Arithmetic mean Arithmetic mean value is called the mean value of the sign, in the calculation of the total volume of which feature in the aggregate remains unchanged

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Characteristics of the arithmetic mean The arithmetic mean has a number of mathematical properties that can be used to calculate it in a simplified way. 1. If the data values (Xi) to reduce or increase by a constant number (A), the mean, respectively, decrease or increase by a same constant number (A)

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2. If the data values (Xi) divided or multiplied by a constant number (A), the mean decrease or increase, respectively, in the same amount of time (this feature allows you to change the frequency of specific gravities - relative frequency): 2. If the data values (Xi) divided or multiplied by a constant number (A), the mean decrease or increase, respectively, in the same amount of time (this feature allows you to change the frequency of specific gravities - relative frequency): a) when divided by a constant number: b) when multiplied by a constant number:

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3. If the frequency divided by a constant number, the mean will not change: 3. If the frequency divided by a constant number, the mean will not change:

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4. Multiplying the mean for the amount of frequency equal to the sum of multiplications variants on the frequency: 4. Multiplying the mean for the amount of frequency equal to the sum of multiplications variants on the frequency: If then the following equality holds:

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5.The sum of the deviations of the number in a data value from the mean is zero: 5.The sum of the deviations of the number in a data value from the mean is zero: If then So

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Measures of location for ungrouped data In calculating summary values for a data collection, the best is to find a central, or typical, value for the data. More important measures of central tendency are presented in this section: Mean (simple or weighter) Median and Mode

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Example - The sales of the six largest restaurant chains are presented in table

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MEDIAN for ungrouped data The median of a data is the middle item in a set of observation that are arranged in order of magnitude. The median is the measure of location most often reported for annual income and property value data. A few extremely large incomes or property values can inflate the mean.

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Position of median If n is odd: Median item number = (n+1)/2 If n is even: Calculate (n+1)/2 The median is the average of the values before and after (n+1)/2.

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Example The median number of people treated daily at the emergency room of St. Luke’s Hospital must be determined from the following data for the last six days: 25, 26, 45, 52, 65, 78 Since the data values are arranged from lowest to highest, the median be easily found. If the data values are arranged in a mess, they must rank. Median item number = (6+1)/2 =3,5 Since the median is item 3,5 in the array, the third and fourth elements need to be averaged: (45+52)/2=48,5. Therefore, 48,5 is the median number of patients treated in hospital emergency room during the six-day period.

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Is used if М = const: Is used if М = const: Harmonic mean is also called the simple mean of the inverse values .

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For example: One student spends on a solution of task 1/3 hours, the second student – ¼ (quarter) and the third student 1/5 hours. Harmonic mean will be calculated:

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Geometric mean for ungrouped data This value is used as the average of the relations between the two values, or in the ranks of the distributions presented in the form of a geometric progression.

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Where П – the multiplication of the data value (Xi). n – power of root

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Example

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Average seniority employee is: Average seniority employee is:

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Harmonic mean for grouped data Harmonic mean - is the reciprocal of the arithmetic mean. Harmonic mean is used when statistical information does not contain frequencies, and presented as xf = M.

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Harmonic mean for grouped data Harmonic mean is calculated by the formula: where M = xf

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Example

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is calculated by the formula: is calculated by the formula: Where fi – frequency of the data value (Xi) П – multiplication sign.

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Mode is calculated by the formula: Mode is calculated by the formula: where хМо – lower boundary of the modal interval i= хМо – xMo+1 - difference between the lower boundary of the modal interval and upper boundary fMo, fMo-1, fMo+1 – frequencies of the modal interval, of interval foregoing modal interval and of interval following modal interval

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We substitute the data into the formula: We substitute the data into the formula: Mo = 12,3 So, the most frequent number of calls per hour = 12.3

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Relationship between mean, median, and mode If a distribution is symmetrical: the mean, median and mode are the same and lie at centre of distribution

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EXAMPLE Consider a study of the hourly wage rates in three different companies, For simplicity, assume that they employ the same number of employees: 100 people.

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So we have three 100-element samples, which have the same average value (35) and the same variability (120). But these are different samples. The diversity of these samples can be seen even better when we draw their histograms. So we have three 100-element samples, which have the same average value (35) and the same variability (120). But these are different samples. The diversity of these samples can be seen even better when we draw their histograms.

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The histogram for company I (left chart) is symmetric. The histogram for company II (middle chart) is right skewed. The histogram for company III (right chart) is left skewed. It remains for us to find a way of determining the type of asymmetry (skewness) and “distinguishing” it from symmetry. The histogram for company I (left chart) is symmetric. The histogram for company II (middle chart) is right skewed. The histogram for company III (right chart) is left skewed. It remains for us to find a way of determining the type of asymmetry (skewness) and “distinguishing” it from symmetry.

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Knowing the median, modal and average values enables us to resolve the problem regarding the symmetry of the distribution of the sample. Hence, Knowing the median, modal and average values enables us to resolve the problem regarding the symmetry of the distribution of the sample. Hence, For symmetrical distributions: x = Me = Mo , For right skewed distributions: x > Me > Mo For left skewed distributions: x < Me < Mo .

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We obtain the following relevant indicators (measures) of asymmetry: We obtain the following relevant indicators (measures) of asymmetry: Index of skewness: ; Standardized skewness ratio: Coefficient of asymmetry