In addition to using different types of text formatting such as changing the font, using bold or italics, sometimes it is necessary to make an underline in Word. Placing a line over a letter is quite simple; let’s look at several ways to solve this problem.

Using "Diacritics"

Thanks to the symbol panel, you can make a dash on top as follows. Position the mouse cursor at the desired location in the text. Go to the “Insert” tab, then find and click in the “Symbols” area on the “Formula” button and select “Insert new formula” from the drop-down menu.

An additional tab “Working with Formulas” or “Designer” will open. From the options presented, in the “Structures” area, select “Diacritics” and click on the window called “Stroke”.

In the added window, type the required word or letter.

The result will look like this.

Emphasizing from above using a figure

Using shapes in Word, you can underline a word both above and below. Consider the underscore. Initially, you need to print the desired text. Next, go to the “Insert” tab in the “Illustrations” area and select the “Shapes” button. In the new window, click on the “Line” shape.

Place a cross over the word at the beginning, press and drag the line to the end of the word, moving up or down, align the line and release.

You can change the color of the upper underline by clicking on the line and opening the “Format” tab. By clicking on the “Shape Outline” button, select the desired color. You can also change the underline type and thickness. To do this, go to the sub-item below “Thickness” or “Strokes”.

In accordance with the settings, the stick can be converted into a dash-dotted line, or changed into an arrow in the desired direction.

Thanks to such simple options, it won’t take much time to put a line over a letter or number. You just have to choose the most suitable method from the above.

In the top row of the formula editor toolbar there are buttons for inserting more than 150 mathematical symbols into a formula. To insert a symbol into a formula, click the button in the top row of the toolbar, and then select a specific symbol from the palette below the button.

In the bottom row of the formula editor toolbar there are buttons for inserting patterns or structures that include symbols such as fractions, radicals, sums, integrals, products, matrices or various parentheses, or corresponding pairs of symbols such as parentheses and square brackets. Many templates contain special fields for entering text and inserting symbols. The formula editor has about 120 templates grouped into palettes. Templates can be nested one inside another to build complex multi-step formulas.

Inserting math symbols into a formula

To insert mathematical symbols into a formula, use the top row of buttons on the formula editor toolbar. Using these buttons, you can insert more than 150 mathematical symbols into your formula.

Table 1

	Inserting relation symbols into a formula
	Inserting spaces and ellipses in a formula
	Add superscripts to a formula
	Inserting Operators in a Formula
	Inserting arrows in a formula
	Inserting logical symbols into a formula
	Inserting set theory symbols into a formula
	Inserting different characters into a formula
	Inserting Greek letters into a formula

Inserting a math template into a formula

The buttons in the bottom row of the Formula Editor toolbar are for inserting mathematical patterns into a formula, such as fractions, radicals, sums, integrals, products, and various types of parentheses.

table 2

	Inserting delimiter patterns into a formula
	Inserting Fraction and Radical Patterns into a Formula
	Creating superscripts and subscripts in a formula
	Creating amounts in a formula
	Inserting an integral into a formula
	Creating mathematical expressions with overbars and underbars
	Create arrows with text in a formula
	Inserting products and set theory patterns into a formula
	Inserting matrix templates into a formula

Task A

To the right of the samples, type the following formulas:

Space characters

The SPACEBAR key does not work in the formula editor because the required spacing between characters occurs automatically. If the need to enter a space nevertheless arises, they can be entered using the Spaces and Ellipses button on the Formula toolbar (see Table 1).

Using the space characters, you can insert five sizes of spaces into a formula. They serve to change automatically set intervals.

If there is a need to change the intervals when entering a formula, you should place the cursor at the place where the interval is changed, and then select one of the symbols of the “Spaces and Ellipses” palette shown in Table 3.

Table 3

Symbol	Description
	Zero space
	Space 1 pt
	Short space (one sixth of a long space)
	Middle space (one third of a long space)
	Long space

Alignment symbol

There is an alignment symbol in the Spaces and Ellipses button palette. This symbol aligns multiple lines in a stack of formulas. Place a character on each line where you want it to be aligned. The lines will be shifted so that the alignment characters are stacked on top of each other.

Alignment symbols are displayed on the screen only in the Equation Editor window. They are not visible in the document and are not printed.

Task B

Try to understand the technology of using the Spaces and Ellipses button on your own using the example of entering the following formulas (enter your formulas in the table below the example):

Clue

After the sum sign, enter a long space using the Spaces and ellipses button at the top of the formula editor toolbar. After the parentheses, enter a middle space.

Align both formulas with the equal sign.

Note. To align formulas with an equal sign, you can select them and then choose Align with = from the Format menu.

Ellipsis symbols

An ellipsis indicates the omission of elements that can usually be easily reconstructed from the context. In the formula editor, there are horizontal, vertical, and diagonal ellipses that can be used when appropriate.

It is advisable to use ellipses when creating vectors and matrices, for example, when creating a general matrix.

In such a matrix, you can enter a 4*4 matrix template in parentheses and fill its fields with alignment symbols and corresponding ellipsis symbols (Fig. 3).

Task B

To the right of the samples, type the following matrices:

Dimensions of formula elements

In the Formula Editor, the size of a symbol is determined by its purpose in the formula, such as whether the symbol is a subscript or exponent symbol.

Each field in the formula corresponds to a certain size. When a character is entered into a field, it takes on the size of the field.

Using standard dimension types to design formula elements

The size of the symbol in the formula can be changed to any of the standard sizes, or you can set the exact size of the symbol, sequence of symbols, or pattern symbol in points.

Selection of standard size type:

Select the required elements.

Choose from five standard sizes from the Size menu. The values ​​of standard sizes can be viewed by selecting the command Size – Define (Figure 4).

Note. On the right side of this command window is an example of the selected symbol. By selecting one of the standard sizes in the Size command window, you can use the sample to immediately determine what type of symbols it will be applied to.

Direct size setting:

Select the formula to edit.

Select the required elements.

Select Custom from the Size menu.

In the Size field, enter the size of the element in points (from 2 to 127). (At one point - 0.352 mm.)

Click OK.

Task B

In the formula below, set the size of the main characters to 20 pt, the size of the subscript/superscript characters to 12 pt. For this:

Double-click to highlight the formula to edit.

Select the desired symbol or group of symbols.

Select Size - Custom.

In the window that appears, specify the desired size.

Click OK to accept your changes.

Changing Standard Dimension Types

By changing the definition of a size type, you can quickly select the size of all characters of a specified type. To redefine standard size types, use the menu command Size - Define.

By default, the size is specified in points. To change the unit of measurement, add one of the abbreviations given in Table 4 to the number.

Table 4

To preview the changes you make, click Apply. To restore the previous dimensions, click Default. To accept the changes, click OK.

Changes made in the Dimensions window will only be reflected in the open formula. They will be taken into account in the formulas of other documents only if these formulas are changed.

Task B

Enter the following formula:

Edit by setting the following character sizes:

regular characters – 16 pt;

large index – 9 pt;

large symbol – 24 pt

For this:

Double-click to select the formulas to edit.

To change size types, select the menu command Size – Set.

Change the size types of the desired symbols.

After previewing (Apply button), click OK to accept your changes.

Questions for control

What operations is the MicrosoftEquation formula editor designed to perform?

Can I use the Microsoft Equation editor to perform calculations?

What is the top row of the MicrosoftEquation toolbar? Bottom row?

When you enter a formula, you can enter part of the formula without using the MicrosoftEquation editor. Should this method be preferred? Why?

Is it possible to change the size of an individual character? Categories? Is it possible to change the default character size?

If you have completed all the tasks and are ready to answer the questions from the list above, then invite the teacher and show him everything that you have created. Be prepared for him to ask you something.

Let X 1, X 2 ... X n- sample of independent random variables.

Let's order these values ​​in ascending order, in other words, build a variation series:

X (1)< Х (2) < ... < X (n) , (*)

Where X (1) = min (X 1, X 2 ... X n),

X (n) = max (X 1, X 2 ... X n).

The elements of a variation series (*) are called ordinal statistics.

Quantities d (i) = X (i+1) - X (i) are called spacings or distances between order statistics.

In scope sample is called the quantity

R = X(n) - X(1)

In other words, the range is the distance between the maximum and minimum members of the variation series.

Sample mean equals: = (X 1 + X 2 + ... + X n) / n

Average

Most of you have probably used important descriptive statistics such as average.

Average is a very informative measure of the “centrality” of an observed variable, especially if its confidence interval is reported. The researcher needs statistics that allow him to draw conclusions about the population as a whole. One such statistic is the average.

Confidence interval for the mean represents the interval of values ​​around the estimate where, with a given level of confidence, the “true” (unknown) population mean lies.

For example, if the sample mean is 23, and the lower and upper limits of the confidence interval with the level p=.95 are 19 and 27, respectively, then we can conclude that with a 95% probability the interval with boundaries 19 and 27 covers the population average.

If you set a higher confidence level, the interval becomes wider, so the probability with which it “covers” the unknown population mean increases, and vice versa.

It is well known, for example, that the more “uncertain” a weather forecast is (i.e., the wider the confidence interval), the more likely it is to be correct. Note that the width of the confidence interval depends on the volume or size of the sample, as well as on the spread (variability) of the data. Increasing the sample size makes the estimate of the mean more reliable. Increasing the spread of observed values ​​reduces the reliability of the estimate.

The calculation of confidence intervals is based on the assumption of normality of the observed values. If this assumption is not met, the estimate may be poor, especially for small samples.

As the sample size increases, say to 100 or more, the quality of the estimate improves without assuming sample normality.

It is quite difficult to “feel” numerical measurements until the data is meaningfully summarized. A diagram is often useful as a starting point. We can also compress information using important characteristics of the data. In particular, if we knew what the represented quantity consisted of, or if we knew how widely dispersed the observations were, then we could form an image of the data.

The arithmetic mean, often simply called the "mean", is obtained by adding all the values ​​and dividing that sum by the number of values ​​in the set.

This can be shown using an algebraic formula. Kit n observations of a variable X can be depicted as X 1, X 2, X 3, ..., X n. For example, for X we can indicate the height of the individual (cm), X 1 denotes growth 1 -th individual, and X i- height i-th individual. The formula for determining the arithmetic mean of observations (pronounced “X with a line”):

= (X 1 + X 2 + ... + X n) / n

You can shorten this expression:

where (the Greek letter "sigma") means "summation", and the indices below and above this letter mean that the summation is made from i = 1 before i = n. This expression is often shortened even further:

Median

If you order data by value, starting with the smallest value and ending with the largest, then the median will also be the averaging characteristic of the ordered set of data.

Median divides a series of ordered values ​​in half with an equal number of those values ​​both above and below it (to the left and right of the median on the number axis).

It is easy to calculate the median if the number of observations n odd. This will be an observation number (n+1)/2 in our ordered data set.

For example, if n=11, then the median is (11 + 1)/2 , i.e. 6th observation in an ordered data set.

If n even, then, strictly speaking, there is no median. However, we usually calculate it as the arithmetic mean of two adjacent means of observations in an ordered data set (i.e., observations number (n/2) And (n/2 + 1)).

So, for example, if n = 20, then the median is the arithmetic mean of observations number 20/2 = 10 And (20/2 + 1) = 11 in an ordered data set.

Fashion

Fashion is the value that occurs most frequently in the data set; if the data is continuous, then we usually group it and calculate the modal group.

Some data sets have no mode because each value occurs only 1 time. Sometimes there is more than one mode; this occurs when 2 or more values ​​occur the same number of times and the occurrence of each of these values ​​is greater than that of any other value.

Fashion is rarely used as a generalizing characteristic.

Geometric mean

If the data distribution is asymmetric, the arithmetic mean will not be a general indicator of the distribution.

If the data is skewed to the right, you can create a more symmetric distribution by taking the logarithm (base 10 or base e) of each variable value in the data set. The arithmetic mean of the values ​​of these logarithms is a characteristic of the distribution for the transformed data.

To obtain a measure with the same units as the original observations, it is necessary to carry out the inverse transformation - potentiation (i.e., take the antilogarithm) of the average logarithm of the data; we call this quantity geometric mean.

If the distribution of log data is approximately symmetric, then the geometric mean is similar to the median and less than the mean of the raw data.

Weighted average

Weighted average used when some values ​​of the variable we are interested in x more important than others. We add weight w i to each of the values x i in our sample to account for this importance.

If the values x 1 , x 2 ... x n have the appropriate weight w 1, w 2 ... w n, then the weighted arithmetic average looks like this:

For example, suppose we are interested in determining the average length of hospitalization in an area and know the average recovery period of patients in each hospital. We take into account the amount of information, taking as a first approximation the number of patients in the hospital as the weight of each observation.

A weighted average and an arithmetic average are identical if each weight is equal to one.

Range (change interval)

Scope is the difference between the maximum and minimum values ​​of the variable in the data set; these two quantities denote their difference. Note that the range is misleading if one of the values ​​is an outlier (see Section 3).

Range derived from percentiles

What are percentiles

Suppose we arrange our data in order from the smallest value of the variable X and up to the largest value. Magnitude X, up to which 1% of observations are located (and above which 99% of observations are located) is called first percentile.

Magnitude X, to which 2% of observations are located is called 2nd percentile, etc.

Quantities X, which divide an ordered set of values ​​into 10 equal groups, i.e. 10th, 20th, 30th,..., 90th and percentiles, are called deciles. Quantities X, which divide the ordered set of values ​​into 4 equal groups, i.e. The 25th, 50th and 75th percentiles are called quartiles. The 50th percentile is median.

Applying percentiles

We can achieve a form of describing scattering that is not affected by an outlier (an anomalous value) by eliminating extreme values ​​and determining the magnitude of the remaining observations.

The interquartile range is the difference between the 1st and 3rd quartiles, i.e. between the 25th and 75th percentiles. It consists of the center 50% of observations in an ordered set, with 25% of the observations below the center point and 25% above it.

The interdecile range contains the central 80% of observations, that is, those observations that lie between the 10th and 90th percentiles.

We often use the range, which contains 95% of the observations, i.e. it excludes 2.5% of observations from below and 2.5% from above. Indication of such an interval is relevant, for example, for diagnosing a disease. This interval is called reference interval, reference range or normal span.

Dispersion

One way to measure the dispersion of data is to determine the degree to which each observation deviates from the arithmetic mean. Obviously, the greater the deviation, the greater the variability, variability of observations.

However, we cannot use the average of these deviations as a measure of dispersion, because positive deviations compensate for negative deviations (their sum is zero). To solve this problem, we square each deviation and find the average of the squared deviations; this quantity is called variation, or dispersion.

Let's take n observationsx 1 , x 2 , x 3 , ..., x n, average which is equal to.

We calculate the variance:

If we are dealing not with a general population, but with a sample, then we calculate sample variance:

Theoretically, it can be shown that a more accurate sample variance will be obtained if one does not divide by n, and on (n-1).

The unit of measurement (dimension) of variation is the square of the units of the original observations.

For example, if measurements are made in kilograms, then the unit of variation will be kilogram squared.

Standard deviation, sample standard deviation

Standard deviation is the positive square root of .

Standard deviation samples is the root of the sample variance.