Lyudmila Prokofievna Kalugina (or simply “Mymra”) in the wonderful film “Office Romance” taught Novoseltsev: “Statistics is a science, it does not tolerate approximation.” In order not to fall under the hot hand of the strict boss Kalugina (and at the same time easily solve tasks from the Unified State Exam and State Examination with elements of statistics), we will try to understand some concepts of statistics that can be useful not only in the thorny path of conquering the Unified State Examination exam, but also simply in everyday life. life.

So what is Statistics and why is it needed? The word “statistics” comes from the Latin word “status”, which means “state and state of affairs”. Statistics deals with the study of the quantitative side of mass social phenomena and processes in numerical form, identifying special patterns. Today, statistics are used in almost all spheres of public life, from fashion, cooking, gardening to astronomy, economics, and medicine.

First of all, when getting acquainted with statistics, it is necessary to study the basic statistical characteristics used for data analysis. Well, let's start with this!

Statistical characteristics

The main statistical characteristics of a data sample (what kind of “sample” is this!? Don’t be alarmed, everything is under control, this incomprehensible word is just for intimidation, in fact, the word “sample” simply means the data that you are going to study) include:

1. sample size,
2. sample range,
3. average,
4. fashion,
5. median,
6. frequency,
7. relative frequency.

Stop, stop, stop! How many new words! Let's talk about everything in order.

Volume and Scope

For example, the table below shows the height of the players of the national football team:

This selection is represented by elements. Thus, the sample size is equal.

The range of the presented sample is cm.

Average

Not very clear? Let's look at our example.

Determine the average height of the players.

Well, shall we get started? We have already figured out that; .

We can immediately safely substitute everything into our formula:

Thus, the average height of a national team player is cm.

Or like this example:

For a week, 9th grade students were asked to solve as many examples from the problem book as possible. The number of examples solved by students per week is given below:

Find the average number of problems solved.

So, in the table we are presented with data on students. Thus, . Well, let’s first find the sum (total number) of all problems solved by twenty students:

Now we can safely begin to calculate the arithmetic mean of the solved problems, knowing that:

Thus, on average, 9th grade students solved each problem.

Here's another example to reinforce.

Example.

On the market, tomatoes are sold by sellers, and prices per kg are distributed as follows (in rubles): . What is the average price of a kilogram of tomatoes on the market?

Solution.

So, what does it equal in this example? That's right: seven sellers offer seven prices, which means ! . Well, we’ve sorted out all the components, now we can start calculating the average price:

Well, did you figure it out? Then do the math yourself average in the following samples:

Answers: .

Mode and median

Let's look again at our example with the national football team:

What is the mode in this example? What is the most common number in this sample? That's right, this is a number, since two players are cm tall; the growth of the remaining players is not repeated. Everything here should be clear and understandable, and the word should be familiar, right?

Let's move on to the median, you should know it from your geometry course. But it’s not difficult for me to remind you that in geometry median(translated from Latin as “middle”) - a segment inside a triangle connecting the vertex of the triangle with the middle of the opposite side. Keyword MIDDLE. If you knew this definition, then it will be easy for you to remember what a median is in statistics.

Well, let's get back to our sample of football players?

Did you notice an important point in the definition of median that we have not yet encountered here? Of course, “if this series is ordered”! Shall we put things in order? In order for there to be order in the series of numbers, you can arrange the height values ​​of football players in both descending and ascending order. It is more convenient for me to arrange this series in ascending order (from smallest to largest). Here's what I got:

So, the series has been sorted, what other important point is there in determining the median? That's right, an even and an odd number of members in the sample. Have you noticed that even the definitions are different for even and odd quantities? Yes, you're right, it's hard not to notice. And if so, then we need to decide whether we have an even number of players in our sample or an odd one? That's right - there are an odd number of players! Now we can apply to our sample a less tricky definition of the median for an odd number of members in the sample. We are looking for the number that is in the middle in our ordered series:

Well, we have numbers, which means there are five numbers left at the edges, and height cm will be the median in our sample. Not so difficult, right?

Now let’s look at an example with our desperate children from grade 9, who solved examples during the week:

Are you ready to look for mode and median in this series?

To begin with, let's order this series of numbers (arrange from the smallest number to the largest). The result is a series like this:

Now we can safely determine the fashion in this sample. Which number occurs more often than others? That's right! Thus, fashion in this sample is equal.

We have found the mode, now we can start finding the median. But first, answer me: what is the sample size in question? Did you count? That's right, the sample size is equal. A is an even number. Thus, we apply the definition of median for a series of numbers with an even number of elements. That is, we need to find in our ordered series average two numbers written in the middle. What two numbers are in the middle? That's right, and!

Thus, the median of this series will be average numbers and:

- median the sample under consideration.

Frequency and relative frequency

That is frequency determines how often a particular value is repeated in a sample.

Let's look at our example with football players. We have before us this ordered series:

Frequency is the number of repetitions of any parameter value. In our case, it can be considered like this. How many players are tall? That's right, one player. Thus, the frequency of meeting a player with height in our sample is equal. How many players are tall? Yes, again one player. The frequency of meeting a player with height in our sample is equal. By asking and answering these questions, you can create a table like this:

Well, everything is quite simple. Remember that the sum of the frequencies must equal the number of elements in the sample (sample size). That is, in our example:

Let's move on to the next characteristic - relative frequency.

Let us turn again to our example with football players. We have calculated the frequencies for each value; we also know the total amount of data in the series. We calculate the relative frequency for each growth value and get this table:

Now create tables of frequencies and relative frequencies yourself for an example with 9th graders solving problems.

Graphical representation of data

Very often, for clarity, data is presented in the form of charts/graphs. Let's look at the main ones:

1. bar chart,
2. pie chart,
3. bar chart,
4. polygon

Column chart

Column charts are used when they want to show the dynamics of changes in data over time or the distribution of data obtained as a result of a statistical study.

For example, we have the following data on the grades of a written test in one class:

The number of people who received such an assessment is what we have frequency. Knowing this, we can make a table like this:

Now we can build visual bar graphs based on such an indicator as frequency(the horizontal axis shows the grades; the vertical axis shows the number of students who received the corresponding grades):

Or we can construct a corresponding bar graph based on the relative frequency:

Let's consider an example of the type of task B3 from the Unified State Examination.

Example.

The diagram shows the distribution of oil production in countries around the world (in tons) for 2011. Among the countries, the first place in oil production was occupied by Saudi Arabia, the United Arab Emirates took seventh place. Where did the USA rank?

Answer: third.

Pie chart

To visually depict the relationship between parts of the sample under study, it is convenient to use pie charts.

Using our table with the relative frequencies of the distribution of grades in the class, we can construct a pie chart by dividing the circle into sectors proportional to the relative frequencies.

A pie chart retains its clarity and expressiveness only with a small number of parts of the population. In our case, there are four such parts (in accordance with possible estimates), so the use of this type of diagram is quite effective.

Let's look at an example of the type of task 18 from the State Examination Inspectorate.

Example.

The diagram shows the distribution of family expenses during a seaside holiday. Determine what the family spent the most on?

Answer: accommodation.

Polygon

The dynamics of changes in statistical data over time are often depicted using a polygon. To construct a polygon, points are marked in the coordinate plane, the abscissas of which are moments in time, and the ordinates are the corresponding statistical data. By connecting these points successively with segments, a broken line is obtained, which is called a polygon.

Here, for example, we are given the average monthly air temperatures in Moscow.

Let's make the given data more visual - we'll build a polygon.

The horizontal axis shows the months, and the vertical axis shows the temperature. We build the corresponding points and connect them. Here's what happened:

Agree, it immediately became clearer!

A polygon is also used to visually depict the distribution of data obtained as a result of a statistical study.

Here is the constructed polygon based on our example with the distribution of scores:

Let's consider a typical task B3 from the Unified State Examination.

Example.

In the figure, bold dots show the price of aluminum at the close of exchange trading on all working days from August to August of the year. The dates of the month are indicated horizontally, and the price of a ton of aluminum in US dollars is indicated vertically. For clarity, the bold points in the figure are connected by a line. Determine from the figure what date the aluminum price at the close of trading was the lowest for the given period.

Answer: .

bar chart

Interval data series are depicted using a histogram. A histogram is a stepped figure made up of closed rectangles. The base of each rectangle is equal to the length of the interval, and the height is equal to the frequency or relative frequency. Thus, in a histogram, unlike a regular bar chart, the bases of the rectangle are not chosen arbitrarily, but are strictly determined by the length of the interval.

For example, we have the following data on the growth of players called up to the national team:

So we are given frequency(number of players with corresponding height). We can complete the table by calculating the relative frequency:

Well, now we can build histograms. First, let's build based on frequency. Here's what happened:

And now, based on the relative frequency data:

Example.

Representatives of companies came to the exhibition on innovative technologies. The chart shows the distribution of these companies by number of employees. The horizontal line represents the number of employees in the company, the vertical line shows the number of companies with a given number of employees.

What percentage are companies with a total number of employees of more than one person?

Answer: .

Brief summary

Sample size- the number of elements in the sample.

Sample range- the difference between the maximum and minimum values ​​of the sample elements.

Arithmetic mean of a series of numbers is the quotient of dividing the sum of these numbers by their number (sample size).

Mode of number series- the number most often found in a given series.

Medianordered series of numbers with an odd number of terms- the number that will be in the middle.

Median of an ordered series of numbers with an even number of terms- the arithmetic mean of two numbers written in the middle.

Frequency- the number of repetitions of a certain parameter value in the sample.

Relative frequency

For clarity, it is convenient to present data in the form of appropriate charts/graphs

• ELEMENTS OF STATISTICS. BRIEFLY ABOUT THE MAIN THINGS.

Statistical sampling- a specific number of objects selected from the total number of objects for research.

Sample size is the number of elements included in the sample.

Sample range is the difference between the maximum and minimum values ​​of sample elements.

Or, sample range

Average of a series of numbers is the quotient of dividing the sum of these numbers by their number

The mode of a series of numbers is the number that appears most frequently in a given series.

The median of a series of numbers with an even number of terms is the arithmetic mean of the two numbers written in the middle, if this series is ordered.

Frequency represents the number of repetitions, how many times over a certain period a certain event occurred, a certain property of an object manifested itself, or an observed parameter reached a given value.

Relative frequency is the ratio of frequency to the total number of data in the series.

Well, the topic is over. If you are reading these lines, it means you are very cool.

Because only 5% of people are able to master something on their own. And if you read to the end, then you are in this 5%!

Now the most important thing.

You have understood the theory on this topic. And, I repeat, this... this is just super! You are already better than the vast majority of your peers.

The problem is that this may not be enough...

For what?

For successfully passing the Unified State Exam, for entering college on a budget and, MOST IMPORTANTLY, for life.

I won’t convince you of anything, I’ll just say one thing...

People who have received a good education earn much more than those who have not received it. This is statistics.

But this is not the main thing.

The main thing is that they are MORE HAPPY (there are such studies). Perhaps because many more opportunities open up before them and life becomes brighter? Don't know...

But think for yourself...

What does it take to be sure to be better than others on the Unified State Exam and ultimately be... happier?

GAIN YOUR HAND BY SOLVING PROBLEMS ON THIS TOPIC.

You won't be asked for theory during the exam.

You will need solve problems against time.

And, if you haven’t solved them (A LOT!), you’ll definitely make a stupid mistake somewhere or simply won’t have time.

It's like in sports - you need to repeat it many times to win for sure.

Find the collection wherever you want, necessarily with solutions, detailed analysis and decide, decide, decide!

You can use our tasks (optional) and we, of course, recommend them.

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In conclusion...

If you don't like our tasks, find others. Just don't stop at theory.

“Understood” and “I can solve” are completely different skills. You need both.

Find problems and solve them!

It is used to visually depict the distribution of specific parameter values ​​by repetition frequency over a certain period of time. It can be used when plotting acceptable values. You can determine how often it falls within or outside the acceptable range. The procedure for constructing a histogram:

1. conduct observations of a random variable and determine its numerical values. The number of experimental points must be at least 30

2. determine the range of the random variable, it determines the width of the histogram R and is equal to Xmax – Xmin

3. the resulting range is divided into k intervals, interval width h = R/k.

4. distribute the received data into intervals - the boundaries of the first interval, - the boundaries of the last interval. Determine the number of points falling in each interval.

5. Based on the received data, a histogram is built. Frequencies are plotted along the ordinate axis, and interval boundaries are plotted along the abscissa axis.

6. Based on the shape of the resulting histogram, they find out the state of the batch of products, the technological process and make management decisions.

Typical types of histograms:

1) Typical or (symmetrical). This histogram indicates the stability of the process

2) Multimodal view or comb. Such a histogram indicates the instability of the process.

3) Distribution with a break on the left or right

4) Plateau (uniform rectangular distribution, such a histogram is obtained in the case of combining several distributions in which the average values ​​differ slightly) analyze such a histogram using the stratification method

5) Two-peak (bimodal) - here two symmetrical ones are mixed with distant average values ​​(tops). Stratification is carried out according to 2 factors. This histogram indicates the occurrence of measurement error

6) With an isolated peak - this histogram indicates the occurrence of a measurement error

Pareto chart.

(20% of people – 80% of income)

In 1887, V. Pareto came up with a formula according to which 80% of money belongs to 20% of people.

In the 20th century, Joseph Juran used this principle to classify quality problems into those that are few but significant and those that are numerous but not significant. According to this method, the vast majority of defects and associated losses arise from a relatively small number of causes.

The Pareto chart is a tool that allows you to distribute efforts to resolve emerging problems and identify the root causes that need to be analyzed first. Constructing a Pareto chart:

1) Defining the goal. The data collection period is set

2) Organization and conduct of observations. A checklist for data recording is developed

3) Analysis of observation results, identification of the most significant factors. A special table form for the data is being developed. The data are arranged in order of importance for each factor. The last row of the table is always the “other factors” group

4) Constructing a Pareto chart

Example: Pareto chart for analyzing types of defects of any product.

To account for the cumulative percentage of losses from several defects, a cumulative curve is constructed.

Analyzing the diagram: When constructing a diagram, you need to pay attention to:

1) it is more effective if the number of factors is more than 10

2) if “other” is too large, you should repeat the analysis of its contents and re-analyze everything

3) if the factor that comes first is difficult to analyze, you should start the analysis with the next one

4) if a factor is discovered that is easy to improve, then this should be taken advantage of, regardless of the order of the factors

5) stratification by factors when processing data

Control cards

They allow you to monitor the progress of the process and influence it using feedback, preventing deviations from the requirements presented to the process. Any map has 3 lines:

1) central line - shows the required average value of the characteristics of the controlled parameter K

2), 3) lines of upper and lower control limits - show the maximum permissible limits for changing the value of the controlled parameter

Other names for the method: “Shewhart control charts.”

Any QC, even if initially ineffective, is a necessary means to restore order in process control. For successful implementation of QCs in practice, it is important not only to master the technique of drawing up and maintaining them, but, what is much more important, to learn how to “read” the map correctly. Advantages of the method: indicates the presence of potential problems before the production of defective products begins, improves quality indicators and reduces the cost of ensuring it.

Disadvantages of the method: competent construction of CC is a complex task and requires certain knowledge. The expected result is obtaining objective information for making decisions about the effectiveness of the process.

Management tools

K control tools use primarily numerical data for analysis.

Affinity diagram

A tool that allows you to identify major process violations by combining verbal data. It is built when there are a large number of ideas and they need to be grouped to clarify their connections. Stages:

1) determining the topic of the basis for data collection

2) collecting data during a brainstorming session around the selected topic; data must be collected indiscriminately

3) each message is registered on the card by each participant

4) grouping related data together

Creation principle

common title for A and B

↓ affinity ↓

general heading A general heading B for

for (a) and (c) (c) and (d) ↕

↕ affinity ____________

↓ affinity ↓

oral data (a); oral data (c); oral data (c); oral data (d).

It is used to systematize a large number of associatively related information. The Japanese Union of Scientists and Engineers included the affinity diagram among the seven quality management methods in 1979.

When formulating a topic for discussion, use the “rule of 7 plus or minus 2.” The sentence must have at least 5 and no more than 9 words, including a verb and a noun.

The affinity diagram is used not to work with specific numerical data, but with verbal statements. An affinity diagram should be used mainly when: there is a need to organize a large amount of information (different ideas, different points of view, etc.), the answer or solution is not absolutely obvious to everyone, making a decision requires agreement among team members (and possibly among other stakeholders) to work effectively.

Advantages of the method: reveals the relationship between different pieces of information; the procedure for creating an affinity diagram allows team members to go beyond their usual thinking and contributes to the realization of the team’s creative potential.

Disadvantages of the method: in the presence of a large number of objects (starting from several dozen), the tools of creativity, which are based on human associative abilities, are inferior to the tools of logical analysis.

The Affinity Diagram is the first of the seven quality management techniques that helps develop a more precise understanding of a problem and identifies major process problems by collecting, summarizing, and analyzing a large amount of oral data based on the affinity relationships between each element.

Connection diagram

A tool that allows you to identify logical connections between the main idea and various data.

The purpose of the study using this diagram is to establish connections between the main causes of process disruption, identified using the affinity diagram, and the problems that need to be solved.

Construction: in the center there is an image of the entire problem/task/area of ​​knowledge; thick main branches with captions emanate from the center - they indicate the main sections of the diagram. The main branches further branch into thinner branches. All branches are signed with keywords that make you remember this or that concept. Examples of situations of appropriate use:

1) when the topic is so complex that connections between different ideas cannot be established through normal discussion

2) if the problem can become a prerequisite for a more fundamental new problem

Work on this diagram should be carried out in teams. The initial determination of the final result is very important. Root causes can be generated from an affinity or Ishikawa diagram.

Tree diagram

A tool that provides a systematic determination of the optimal means of solving problems that arise, presented at various levels. Tree diagram structure:

Use cases for the chart:

1) when consumer requirements are unclear regarding the product

2) if it is necessary to investigate all possible elements of the problem

3) at the design stage, when short-term goals must be realized before the result of all the work.

Matrix diagram

A tool that identifies the importance of various connections. Allows you to process a large amount of data with an illustration of logical connections between various elements. The diagram displays the contours of connections and correlations between tasks, functions, characteristics, highlighting their relative importance.

A	IN
	B1	B2	B3	B4	B5	B6
A1			∆
A2						▄0
A3			▄0
A4		▄				▄

A1,..., A4 = components of the objects under study A, B - =//= B

They are characterized by different connection strengths, which are shown using special symbols:

▄0 – strong connection

▄ - medium connection

∆ - weak connection

If there is no shape in a cell, it means there is no connection between the components.

Arrow diagram

An arrow diagram is a tool that allows you to plan the timing of all necessary work for the speedy and successful implementation of your goal. The diagram is widely used in planning and subsequent monitoring of the progress of work. There are 2 types of arrow charts: Gantt chart and network chart. Example of a Gantt chart: building a house within 12 months.

NUMBER

Operation

Months

Foundation

skeleton

Forests

Exterior decoration of the house

Interior

Water pipes

Electrical work

Doors and windows

Painting the interior walls

End of ext. finishing

Final inspection and handover

Example network diagram

A circle with the operation number inside, an arrow to the next circle, below it the number of months. Dotted arrows show the connection of the operation. The stages are the same, except 11 is the final inspection, and 12 is delivery.

A network graph is a graph whose vertices display the states of a certain object (for example, construction), and the arcs represent the work being carried out at this object. Each arc is associated with the time during which the work is carried out and/or the number of workers who carry out the work. Often a network graph is constructed in such a way that the horizontal arrangement of vertices corresponds to the time it takes to reach the state corresponding to a given vertex.

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Histogram (bar graph)

It is used to visually depict the distribution of specific parameter values ​​by repetition frequency over a certain period of time. It can be used when plotting acceptable values. You can determine how often it falls within the acceptable range or goes beyond it. The procedure for constructing a histogram:

• 1. conduct observations of a random variable and determine its numerical values. The number of experimental points must be at least 30
• 2. determine the range of the magnitude, it determines the width of the histogram R and is equal to Xmax - Xmin
• 3. the resulting range is divided into k intervals, interval width h = R/k.
• 4. distribute the received data into intervals - the boundaries of the first interval, - the boundaries of the last interval. Determine the number of points falling in each interval.
• 5. Based on the received data, a histogram is built. Frequencies are plotted along the ordinate axis, and interval boundaries are plotted along the abscissa axis.
• 6. Based on the shape of the resulting histogram, they find out the state of the batch of products, the technological process and make management decisions.

Typical types of histograms:

• 1) Typical or (symmetrical). This histogram indicates the stability of the process
• 2) Multimodal view or comb. Such a histogram indicates the instability of the process.
• 3) Distribution with a break on the left or right
• 4) Plateau (uniform rectangular distribution, such a histogram is obtained in the case of combining several associations, the average values ​​of which differ slightly) analyze such a histogram using the stratification method
• 5) Two-peak (bimodal) - here two symmetrical ones are mixed with distant average values ​​(tops). Stratification is carried out according to 2 factors. This histogram indicates the occurrence of measurement error
• 6) With an isolated peak - this histogram indicates the occurrence of a measurement error

Graphs make it possible to assess the state of the process at the moment, as well as predict a more distant result based on process trends that can be detected. When a graph shows changes in data over time, the graph is also called a time series.

The following types of graphs are usually used: Broken line (line graph), Column and Pie

Line graph

Using a line graph, display the nature of changes in the amount of annual revenue from the sale of products, and also predict the trend in revenue changes in the next two years (we will first do this using the Trend function).

Revenue, thousand USD

Create a new Excel workbook. We enter the title of the work, as well as the initial data, after which we build a line graph. We edit the resulting diagram using context menus.

The nature of changes in revenue, as well as the forecast, is given by a trend line, which can be constructed by opening the context menu on the broken line and selecting the command Add a trend line .

In the dialog box that opens, on the tab Type Possible types of trend line are shown. To select the line type that best fits the data, you can do the following: place trend lines of each acceptable type in order on the chart (i.e., linear, logarithmic, second-degree polynomial, power, and exponential), specifying for each line on the tab Options forecast ahead by 1 unit (year) and placement on the diagram of the approximation reliability value. Moreover, after constructing the next line, the value of the reliability of the approximation R 2 (The most reliable trend line is the one for which the value of R 2 is equal to or close to one).

The greatest reliability of the approximation is provided by a polynomial line with degree two (R 2 = 0.6738), which we choose as the trend line. To do this, we remove all trend lines from the diagram, after which we restore a polynomial line of the second degree.

Using the approximating line, we can assume that revenue will tend to increase in the coming year.

Bar graph

A bar graph represents a quantitative relationship expressed by the height of the bar. For example, the dependence of the cost on the type of product, the amount of losses due to defects depending on the process, etc. Typically, bars are shown on a graph in descending order of height from right to left. If among the factors there is a group “Other”, then the corresponding column on the graph is shown on the far right.

The figure shows the results of Table 1 above in the form of a bar graph.

Circular graph.

A circular graph expresses the ratio of the components of an entire parameter, for example, the ratio of the amounts of revenue from sales separately by type of part and the total amount of revenue; ratio of elements that make up the cost of the product, etc.

In Fig. The ratio of combine failures by components and assemblies is shown in the form of a circular graph.

	Type of failure	Number of failures
	Harvest part
	Hydraulic equipment
	Thresher
	Electrical equipment
	hydraulic transmission

During this lesson we will become familiar with bar charts and learn how to use them. Let's determine in which cases it is more convenient to use pie charts and in which it is more convenient to use column charts. Let's learn how to apply diagrams in real life.

Rice. 1. Pie chart of ocean areas versus total ocean area

In Figure 1 we see that the Pacific Ocean is not only the largest, but also occupies almost exactly half of the entire world's oceans.

Let's look at another example.

The four closest planets to the Sun are called terrestrial planets.

Let's write down the distance from the Sun to each of them.

Mercury is 58 million km away

Venus is 108 million km away

150 million km to Earth

Mars is 228 million km away

We can again create a pie chart. It will show how much the distance for each planet contributes to the sum of all distances. But the sum of all distances does not make sense to us. A full circle does not correspond to any value (see Fig. 2).

Rice. 2 Pie chart of distances to the Sun

Since the sum of all quantities does not make sense to us, there is no point in constructing a pie chart.

But we can depict all these distances using the simplest geometric shapes - rectangles or columns. Each value will have its own column. How many times greater is the value, the higher is the column. We are not interested in the sum of quantities.

To make it easier to see the height of each column, let’s draw a Cartesian coordinate system. On the vertical axis we will mark in millions of kilometers.

And now we will build 4 columns with a height corresponding to the distance from the Sun to the planet (see Fig. 3).

Mercury is 58 million km away

Venus is 108 million km away

150 million km to Earth

Mars is 228 million km away

Rice. 3. Bar chart of distances to the Sun

Let's compare the two diagrams (see Fig. 4).

A bar chart is more useful here.

1. It immediately shows the shortest and greatest distances.

2. We see that each subsequent distance increases by approximately the same amount - 50 million km.

Rice. 4. Comparison of chart types

Thus, if you are wondering which chart is better for you to build - a pie chart or a column chart, then you need to answer:

Do you need the sum of all quantities? Does it make sense? Do you want to see the contribution of each value to the total, to the sum?

If yes, then you need a circular one, if not, then a columnar one.

The sum of the areas of the oceans makes sense - this is the area of ​​the World Ocean. And we built a pie chart.

The sum of the distances from the Sun to different planets did not make sense to us. And the columnar one turned out to be more useful for us.

Construct a diagram of the change in average temperature for each month throughout the year.

Temperatures are given in Table 1.

September

Table 1

If we add up all the temperatures, the resulting number will not make much sense to us. (It makes sense if we divide it by 12 - we get the average annual temperature, but this is not the topic of our lesson.)

So, let's build a bar chart.

Our minimum value is -18, maximum - 21.

Now let's draw 12 columns for each month.

We draw the columns corresponding to negative temperatures downwards (see Fig. 5).

Rice. 5. Column chart of the change in average temperature for each month during the year

What does this diagram show?

It's easy to see the coldest month and the warmest. You can see the specific temperature value for each month. It can be seen that the warmest summer months differ less from each other than the autumn or spring months.

So, to build a bar chart, you need:

1) Draw coordinate axes.

2) Look at the minimum and maximum values ​​and mark the vertical axis.

3) Draw bars for each value.

Let's see what surprises may arise during construction.

Construct a bar graph of the distances from the Sun to the nearest 4 planets and the nearest star.

We already know about the planets, and the nearest star is Proxima Centauri (see Table 2).

Table 2

All distances are again in millions of kilometers.

We build a bar chart (see Fig. 6).

Rice. 6. Bar chart of the distance from the sun to the terrestrial planets and the nearest star

But the distance to the star is so enormous that against its background the distances to the four planets become indistinguishable.

The diagram has lost all meaning.

The conclusion is this: you cannot build a chart based on data that differs from each other by a thousand or more times.

So what to do?

You need to split the data into groups. For planets, construct one diagram, as we did, for stars, another.

Construct a bar chart for the melting temperatures of metals (see Table 3).

Table 3. Melting temperatures of metals

If we build a diagram, we hardly see the difference between copper and gold (see Fig. 7).

Rice. 7. Column chart of melting temperatures of metals (graduation from 0 degrees)

All three metals have quite high temperatures. The area of ​​the diagram below 900 degrees is not interesting to us. But then it is better not to depict this area.

Let's start the calibration from 880 degrees (see Fig. 8).

Rice. 8. Column chart of melting temperatures of metals (graduation from 880 degrees)

This allowed us to depict the bars more accurately.

Now we can clearly see these temperatures, as well as which one is higher and by how much. That is, we simply cut off the lower parts of the columns and depicted only the tops, but in approximation.

That is, if all values ​​start from a sufficiently large value, then calibration can begin from this value, and not from zero. Then the diagram will be more visual and useful.

Manual drawing of diagrams is a rather long and labor-intensive task. Today, to quickly make a beautiful chart of any type, you use Excel spreadsheets or similar programs such as Google Docs.

You need to enter the data, and the program itself will build a chart of any type.

Let's build a diagram illustrating how many people speak which language as their native language.

Data taken from Wikipedia. Let's write them down in an Excel table (see Table 4).

Table 4

Let's select the table with the data. Let's look at the types of diagrams offered.

There are both circular and columnar ones. Let's build both.

Circular (see Fig. 9):

Rice. 9. Pie chart of language shares

Columnar (see Fig. 10)

Rice. 10. A bar chart illustrating how many people speak which language as their mother tongue.

What kind of diagram we need will need to be decided each time. The finished diagram can be copied and pasted into any document.

As you can see, creating diagrams today is not difficult.

Let's see how the diagram helps in real life. Here is information on the number of lessons in basic subjects in sixth grade (see Table 5).

Academic subjects	Number of lessons per week	Number of lessons per year
Russian language
Literature
English language
Mathematics
Story
Social science
Geography
Biology
Music

Table 5

Not very easy to read. Below is a diagram (see Fig. 11).

Rice. 11. Number of lessons per year

And here it is, but the data is arranged in descending order (see Fig. 12).

Rice. 12. Number of lessons per year (descending)

Now we can clearly see which lessons are the most and which are the least. We see that the number of English lessons is two times less than Russian, which is logical, because Russian is our native language and we have to speak, read, and write in it much more often.

Bibliography

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012.
2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - Gymnasium. 2006.
3. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - M.: Education, 1989.
4. Rurukin A.N., Tchaikovsky I.V. Assignments for the mathematics course for grades 5-6. - M.: ZSh MEPhI, 2011.
5. Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A manual for 6th grade students at the MEPhI correspondence school. - M.: ZSh MEPhI, 2011.
6. Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: a textbook-interlocutor for grades 5-6 of secondary school. - M.: Education, Mathematics Teacher Library, 1989.

http://ppt4web.ru/geometrija/stolbchatye-diagrammy0.html

Homework

1. Construct a bar chart of precipitation (mm) per year in Chistopol.

2. Draw a bar graph using the following data.

3. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012. No. 1437.