Guide

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1

**The most common way to estimate the average value in the range of values is the arithmetic mean.To calculate it, you need to sum all the values of a number of**

**divided by the number of these values.For example, if a given number of 3, 4, 8, 12, 17, it is equal to the arithmetic mean (3 + 4 + 8 + 12 + 17) / 5 = 44/5 = 8.6.**

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2

Another medium often encountered in mathematical and statistical problems, called the harmonic mean.The harmonic mean of the numbers a0, a1, a2 ... an equal to n / (1 / a0 + 1 / a1 + 1 / a2 ... + 1 / an).For example, for the same number
ways less than the arithmetic mean.

**, as in the preceding example, the harmonic mean is equal to 5 / (1/3 + 1/4 + 1/8 + 1/12 + 1/17) = 5 / (347/408)= 5.87.The harmonic mean is al** 3

various averages are used in different kinds of tasks.For example, if you know that the first hour of a car driving at the speed of A, and the second - at the speed of B, then the average speed during the journey will be the average between A and B. But if you know that the car traveled at a speed of one kilometer of A,and the next - at a rate of B, then to calculate its average speed during the journey, you will need to take the harmonic mean between A and B.

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4

**For statistical purposes, the arithmetic mean is a convenient and objective evaluation, but onlyin those cases, where the values of a number of**

**no outliers.For example, for**

**number 1, 2, 3, 4, 5, 6, 7, 8, 9, 200 is equal to the arithmetic mean of 24 5 - much greater than all members****series, except the last one.It is obvious that such an estimate can not be considered fully adequate.**

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5

In such cases, we must calculate the median

**number**.This average value, the value of which is exactly in the middle of a number of**so that all members****number**located to the median - it is no more, and all located after - not less.Of course, you must first organize a number of members of**ascending.** 6

If the number a0 ... an odd number of values, that is, n = 2k + 1, the median for

example, already considered a number 1, 2, 3, 4, 5, 6, 7, 8, 9, 200 tenmembers.Consequently, his media - the arithmetic mean between the fifth and sixth members, that is (5 + 6) / 2 = 5.5.This estimate is much better reflects the average value of a typical member of a number

**accepted member****series with the serial number k + 1. If the number of values is even,i.e. n = 2k, then the median of the arithmetic mean is considered members****number with the numbers k and k + 1.**example, already considered a number 1, 2, 3, 4, 5, 6, 7, 8, 9, 200 tenmembers.Consequently, his media - the arithmetic mean between the fifth and sixth members, that is (5 + 6) / 2 = 5.5.This estimate is much better reflects the average value of a typical member of a number

**.**** **

# Tip 2: How to calculate the median

term "median triangle" still occurs in the course of geometry 7th grade, but its presence causes some difficulties and the students graduating from school and their parents.This article describes the method to be compact, so you can find

**median**arbitrary triangle.

you need

- calculator

Guide

1

To start, you should decide on the concept of the median (to know what it means).

Look at arbitrary triangle ABC.BD-piece which connects the tip of the triangle to the midpoint of the opposite side is the median.

Thus, due to the above definition and the accompanying figure 1, you should be clear that every triangle has three medians that intersect within this figure.

point of intersection of the medians of the triangle is the center of gravity, or as it is called, the center of mass.Each median point of intersection of the medians is divided in the ratio 2: 1, counted from the top.

Pay attention to the fact that the triangles, which will be divided into the original triangle, all of its medians have the same area.

Look at arbitrary triangle ABC.BD-piece which connects the tip of the triangle to the midpoint of the opposite side is the median.

Thus, due to the above definition and the accompanying figure 1, you should be clear that every triangle has three medians that intersect within this figure.

point of intersection of the medians of the triangle is the center of gravity, or as it is called, the center of mass.Each median point of intersection of the medians is divided in the ratio 2: 1, counted from the top.

Pay attention to the fact that the triangles, which will be divided into the original triangle, all of its medians have the same area.

2

To calculate the median

where m (a) - the median of triangle ABC, connecting vertex A with the midpoint of the side BUsing,

b - side AC of the triangle ABC,

to - side AB of the triangle ABC,

a - side BC of the triangle ABC.From

represented formula that knowing the lengths of the medians of the triangle, you can find the length of any side of it.

**, you must use a specially designed algorithm.The formula for the calculation of the median through the sides of the triangle appears as shown in Figure 2,**where m (a) - the median of triangle ABC, connecting vertex A with the midpoint of the side BUsing,

b - side AC of the triangle ABC,

to - side AB of the triangle ABC,

a - side BC of the triangle ABC.From

represented formula that knowing the lengths of the medians of the triangle, you can find the length of any side of it.

3

If you need a formula for finding the sides of the triangle through his media, it looks as shown in Figure 3, where:

a - side BC of the triangle ABC,

m (b) - the median leavingAt the apex of,

m (c) - the median, the vertex C,

m (a) -mediana emerging from the top of A.

a - side BC of the triangle ABC,

m (b) - the median leavingAt the apex of,

m (c) - the median, the vertex C,

m (a) -mediana emerging from the top of A.

4

To properly calculate the median, you need to get acquainted with the particular casesthat can occur when solving equations with the presence in them of any triangle.

1. In an equilateral triangle, the median emerging from the top, which is formed by equal sides is:

- bisector of the angle formed by the equal sides of the triangle;

is the height of the triangle;

2. In an equilateral triangle all the medians are equal.All medians are the bisectors of corresponding angles and the height of the triangle.

1. In an equilateral triangle, the median emerging from the top, which is formed by equal sides is:

- bisector of the angle formed by the equal sides of the triangle;

is the height of the triangle;

2. In an equilateral triangle all the medians are equal.All medians are the bisectors of corresponding angles and the height of the triangle.

Sources:

- how to calculate the median number