Descriptive statistics. Average
Objectives: to give concepts, algorithms for finding the arithmetic mean and median, range and mode of a number of numbers, to show the significance of this topic in practical human activity; acquiring practical skills to perform these tasks; increasing the level of mathematical training required by the new standards.
- equip students with a system of knowledge on the topic “Determination of the probability of events, the arithmetic mean and median of a set of numbers”;
- develop skills in applying this knowledge when solving a variety of problems of varying complexity;
- prepare students for passing the State Examination Test;
- develop independent work skills.
During the classes
1. Theoretical part.
1). Finding the probability of events.
IN Everyday life, in practical and scientific activities, certain phenomena are often observed and certain experiments are carried out.
In the process of observation or experiment one encounters some random events, i.e., such events that may or may not happen. For example, getting heads or tails when tossing a coin, hitting a target or missing a shot, winning a sports team in a meeting with an opponent, losing or a draw - all these are random events.
The patterns of random events are studied by a special branch of mathematics called probability theory. Probability theory methods are used in many fields of knowledge.
The origin of probability theory occurred in search of an answer to the question: how often does this or that event occur in a large series of tests occurring under the same conditions with random outcomes.
In order to assess the probability of an event of interest to us, it is necessary to conduct a large number of experiments or observations, and only after that can the probability of this event be determined.
For example, throwing a die. When throwing a die, the chances of each number from 1 to 6 appearing on its top face are the same. They say there are 6 equally possible outcomes experience with dice rolling: roll 1,2,3,4,5, and 6 points.
Outcomes in this experiment are considered equally possible if the chances of these outcomes are equal.
Outcomes in which some event occurs are called favorable outcomes for that event.
Definition: the ratio of the number of favorable outcomes N (A) of event A to the number of all equally possible outcomes N of this event is called the probability of event A.
Scheme for finding the probability of an event.
To find the probability of a random event A during a certain test, you should:
- find the number N of all equally possible outcomes of a given test;
- find the number N(A) of those favorable trial outcomes in which event A occurs;
- find the ratio N(A)/N; this is the probability of event A
For example: 1 . A box contains 10 red, 7 yellow and 3 blue balls. What is the probability that a ball taken at random will be yellow?
Solution. Equally possible outcomes - (10+7+3)=20
Favorable outcomes-7
2. There are 5 black balls in the box. What is the smallest number of white balls that must be placed in this box so that the probability of drawing a black ball out of the box at random is no more than 0.15?
Solution: Let x be white balls.
2) Determining and finding the arithmetic mean and median of a series of numbers.
Definition: the arithmetic mean of several numbers is a number equal to the ratio of the sum of these numbers to their number.
The arithmetic mean of a set of numbers x 1 , x 2 , x 3 , x 4 , x 5 is usually denoted as x.
For example, average five numbers will be written like this:
X = (x 1 +x 2 +x 3 +x 4 +x 5)/5
Example: find average rating student in mathematics, if over the past period he received: 3,4,4,5,3,2,4,3.
Solution: (3+4+4+5+3+2+4+3)/8=3.5
Definition: a median is a number that divides a set of numbers into two parts of equal numbers, so that on one side of this number all values are greater than the median, and on the other less. Instead of "median" you could say "middle".
Scheme for finding the median of a set of numbers:
To find the median of a set of numbers:
- arrange a number set (write in ascending order);
- simultaneously cross out the “largest” and “smallest” numbers of a given set of numbers until one number or two numbers remain;
- if one number remains, then it is the median (for an odd set of numbers);
- if there are two numbers left, then the median will be the arithmetic mean of the two remaining numbers (for an even set of numbers).
The median is usually denoted by the letter M.
Example: find the median of a set of numbers: 9,3,1,5,7.
Solution: write the numbers in ascending order: 1,3,5,7,9.
Cross out 1 and 9, 3 and 7. The remaining number 5 is the median. M=5
Example: find the median of a set of numbers 2,3,3,5,7,10.
Solution: cross out 2 and 10, 3 and 7. To find M you need: (3+5)/2= 4. M=4
Determining and finding scope and mode.
Definition: The range of a series of numbers is the difference between the largest and smallest of these numbers.
The range of a series is found when one wants to determine how large the spread of data in a series is.
Definition: The mode of a series of numbers is the number that appears in a given series more often than others.
A series of numbers may have more than one mode, or may not have a mode at all.
Example: In a physical education lesson, 14 schoolchildren were high jumping, and the teacher was recording their results. The result was the following series of data (in cm):
125, 110, 130, 125, 120, 130, 140, 125, 110, 130, 120, 125, 120, 125.
Find the median, range and mode of measurement.
Solution: write down all measurement options in ascending order, separating groups of identical results with spaces:
110, 110, 120, 120, 120, 125, 125, 125, 125, 125, 130, 130, 130, 140.
The measurement range is 140-110=30.
125 - met the greatest number of times, i.e. 5 times; it is a mode of measurement.
2. Practical part.
1). Tasks for independent decision on probability theory.
1. For every 100 light bulbs, on average there are 4 defective ones. What is the probability that a light bulb taken at random will turn out to be working? Answer: 0.96.
2. On average, there are 8 defective CDs per 400 CDs. What is the probability that a CD taken at random will be good? Answer: 0.98.
3. 17 points out of 50 are colored in Blue colour, and 13 of the remaining points are colored orange. What is the probability that a randomly selected point will be colored? Answer: 0.6.
4. One letter is randomly selected from the word “mathematics”. What is the probability that the selected letter occurs only once in this word? Answer: 0.3.
5. One letter is randomly selected from the word “certification”. What is the probability that the chosen letter will be the letter "a"? Answer: 0.2
6. Of the 30 ninth-graders, 4 chose an exam in physics, 12 in social studies, 8 in a foreign language, and the rest in literature. What is the probability that the selected student will take the literature exam. Answer: 0.2.
7. The test in mathematics consists of 15 problems: 4 problems in geometry, 2 problems in probability theory, the rest in algebra. The student made a mistake in one problem. What is the probability that a student made a mistake in an algebra problem? Answer: 0.6.
8. Out of 1000 cars produced in 2007-2009, 150 have a defective brake system. What is the probability of buying a faulty car? Answer: 0.15.
9. Participating in the rhythmic gymnastics competition are: 3 gymnasts from Russia, 3 gymnasts from Ukraine and 4 gymnasts from Belarus. The order of performance will be determined by drawing lots. Find the probability that a gymnast from Russia will compete first. Answer 0.3
10. There are 18 gymnasts performing at the rhythmic gymnastics championship, among them 3 gymnasts from Russia, 2 gymnasts from China. The order of performance is determined by drawing lots. Find the probability that a gymnast from either Russia or China will compete last? Answer: 5/18.
11. From a class of 12 boys and 8 girls, 1 person on duty is chosen by lot. What is the probability that it will be a boy? Answer: 0.6.
12. 2 coins are thrown at the same time. What is the probability of them landing on 2 heads? The answer is 0.25.
2)Problems on finding the arithmetic mean and median, range and mode of a set of numbers.
Milling crews spent on processing one part different time(in min.), presented as a data series: 40; 37; 35; 36; 32; 42; 32; 38; 32. How much does the median of this set differ from the arithmetic mean? Answer: 0.
5 apple tree seedlings were planted in the garden, the height of which in centimeters is as follows: 168, 13, 156, 165, 144. How much does the arithmetic mean of this set of numbers differ from its median? Answer: 3, 8
6 pear trees growing in the garden gave a harvest, the mass of which (in kg) for each of the trees is as follows: 29, 35, 26, 28, 32, 36. How much does the arithmetic mean of this set of numbers differ from its median? Answer: 0.5
The time the cashier served each of several store customers formed the following series of data: 2 minutes. 42 sec., 3 min. 2 sec., 3 imn. 7 sec., 2 min. 54 sec., 2 min. 48 sec. Find the mean and median of this data series. Answer: 2 min. 55 sec., 2 min. 54 sec.
The time between seven calls received by the taxi service formed the following series of data: 34 seconds, 45 seconds, 1 minute. 16 sec., 38 sec., 43 sec., 52 sec. Find the mean and median of this data series. Answer: 48 sec., 44 sec.
Literature : Mordkovich, A. G., I. M. Smirnova. Tutorial for educational institutions(basic level) - M.: Mnemosyne, 2009. - 164 p.
When studying the student workload, a group of 12 seventh graders was identified. They were asked to record the time (in minutes) spent on algebra homework on a given day. We received the following data: 23, 18, 25, 20, 25, 25, 32, 37, 34, 26, 34, 25. When studying the student workload, a group of 12 seventh-graders was identified. They were asked to record the time (in minutes) spent on algebra homework on a given day. We received the following data: 23, 18, 25, 20, 25, 25, 32, 37, 34, 26, 34, 25.
Arithmetic mean of the series. The arithmetic mean of a series of numbers is the quotient of dividing the sum of these numbers by the number of terms. The arithmetic mean of a series of numbers is the quotient of dividing the sum of these numbers by the number of terms.():12=27
Row range. The range of a series is the difference between the largest and smallest of these numbers. The range of a series is the difference between the largest and smallest of these numbers. The largest time consumption is 37 minutes, and the smallest is 18 minutes. Let's find the range of the series: 37 – 18 = 19 (min)
Fashion series. The mode of a series of numbers is the number that appears in a given series more often than others. The mode of a series of numbers is the number that appears in a given series more often than others. The mode of our series is the number - 25. The mode of our series is the number - 25. A series of numbers may or may not have more than one mode. 1) 47,46,50,47,52,49,45,43,53,53,47,52 – two modes 47 and 52. 2) 69,68,66,70,67,71,74,63, 73.72 – no fashion.
The arithmetic mean, range and mode are used in statistics - a science that deals with obtaining, processing and analyzing quantitative data about a variety of mass phenomena occurring in nature and society. The arithmetic mean, range and mode are used in statistics - a science that deals with obtaining, processing and analyzing quantitative data about a variety of mass phenomena occurring in nature and society. Statistics studies the number separate groups population of the country and its regions, production and consumption various types products, transportation of goods and passengers various types transport, Natural resources etc. Statistics studies the size of individual groups of the population of the country and its regions, the production and consumption of various types of products, the transportation of goods and passengers by various modes of transport, natural resources, etc.
1. Find the arithmetic mean and range of a series of numbers: a) 24,22,27,20,16,37; b)30,5,23,5,28, Find the arithmetic mean, range and mode of a number of numbers: a)32,26,18,26,15,21,26; b) -21, -33, -35, -19, -20, -22; b) -21, -33, -35, -19, -20, -22; c) 61,64,64,83,61,71,70; c) 61,64,64,83,61,71,70; d) -4, -6, 0, 4, 0, 6, 8, -12. d) -4,-6, 0, 4, 0, 6, 8, In the series of numbers 3, 8, 15, 30, __, 24 one number is missing. Find it if: a) the arithmetic mean of the series is 18; a) the arithmetic mean of the series is 18; b) the range of the series is 40; b) the range of the series is 40; c) the mode of the series is 24. c) the mode of the series is 24.
4. In the certificate of secondary education, four friends - school graduates - had the following grades: Ilyin: 4,4,5,5,4,4,4,5,5,5,4,4,5,4,4; Ilyin: 4,4,5,5,4,4,4,5,5,5,4,4,5,4,4; Semenov: 3,4,3,3,3,3,4,3,3,3,3,4,4,5,4; Semenov: 3,4,3,3,3,3,4,3,3,3,3,4,4,5,4; Popov: 5,5,5,5,5,4,4,5,5,5,5,5,4,4,4; Popov: 5,5,5,5,5,4,4,5,5,5,5,5,4,4,4; Romanov: 3,3,4,4,4,4,4,3,4,4,4,5,3,4,4. Romanov: 3,3,4,4,4,4,4,3,4,4,4,5,3,4,4. What grade point average did each of these graduates graduate with? Indicate the most typical grade for each of them in the certificate. What statistics did you use to answer? What grade point average did each of these graduates graduate with? Indicate the most typical grade for each of them in the certificate. What statistics did you use to answer?
Independent work Option 1. Option Given a series of numbers: 35, 44, 37, 31, 41, 40, 31, 29. Find the arithmetic mean, range and mode. 2. In the series of numbers 4, 9, 16, 31, _, 25 4, 9, 16, 31, _, 25 one number is missing. one number is missing. Find it if: Find it if: a) the arithmetic mean a) the arithmetic mean is 19; some equals 19; b) range of the series – 41. b) range of the series – 41. Option Given a series of numbers: 38, 42, 36, 45, 48, 45.45, 42. Find the arithmetic mean, range and mode of the range. 2. In the series of numbers 5, 10, 17, 32, _, 26, one number is missing. Find it if: a) the arithmetic mean is 19; b) the range of the series is 41.
The median of an ordered series of numbers with an odd number of numbers is the number written in the middle, and the median of an ordered series of numbers with an even number of numbers is the arithmetic mean of the two numbers written in the middle. The median of an ordered series of numbers with an odd number of numbers is the number written in the middle, and the median of an ordered series of numbers with an even number of numbers is the arithmetic mean of the two numbers written in the middle. The table shows the electricity consumption in January by residents of nine apartments: The table shows the electricity consumption in January by residents of nine apartments: Apartment number Electricity consumption
Let's make an ordered series: 64, 72, 72, 75, 78, 82, 85, 91.93. 64, 72, 72, 75, 78, 82, 85, 91, – median this series. 78 is the median of this series. Given an ordered series: Given an ordered series: 64, 72, 72, 75, 78, 82, 85, 88, 91, 93. ():2 = 80 – median. ():2 = 80 – median.
1. Find the median of a series of numbers: a) 30, 32, 37, 40, 41, 42, 45, 49, 52; a) 30, 32, 37, 40, 41, 42, 45, 49, 52; b) 102, 104, 205, 207, 327, 408, 417; b) 102, 104, 205, 207, 327, 408, 417; c) 16, 18, 20, 22, 24, 26; c) 16, 18, 20, 22, 24, 26; d) 1.2, 1.4, 2.2, 2.6, 3.2, 3.8, 4.4, 5.6. d) 1.2, 1.4, 2.2, 2.6, 3.2, 3.8, 4.4, 5.6. 2. Find the arithmetic mean and median of a series of numbers: a) 27, 29, 23, 31,21,34; a) 27, 29, 23, 31,21,34; b) 56, 58, 64, 66, 62, 74; b) 56, 58, 64, 66, 62, 74; c) 3.8, 7.2, 6.4, 6.8, 7.2; c) 3.8, 7.2, 6.4, 6.8, 7.2; d) 21.6, 37.3, 16.4, 12, 6. d) 21.6, 37.3, 16.4, 12, 6.
3. The table shows the number of exhibition visitors in different days weeks: Find the median of the given data series. On what days of the week was the number of exhibition visitors greater than the median? Days of the week Mon Mon Tue Tue Wed Wed Thu Thu Fri Fri Sat Sun Sun Number of visitors
4. Below is the average daily sugar processing (in thousand centners) by factories sugar industry of a certain region: (in thousand) sugar factories of a certain region: 12.2, 13.2, 13.7, 18.0, 18.6, 12.2, 18.5, 12.4, 12.2 , 13.2, 13.7, 18.0, 18.6, 12.2, 18.5, 12.4, 14, 2, 17.8. 14, 2, 17.8. For the presented series, find the arithmetic mean, mode, range and median. For the presented series, find the arithmetic mean, mode, range and median. 5. The organization kept daily records of letters received during the month. As a result, we received the following data series: 39, 43, 40, 0. 56, 38, 24, 21, 35, 38, 0. 58, 31, 49, 38, 25, 34, 0. 52, 40, 42, 40 , 39, 54, 0, 64, 44, 50, 38, 37, 43, 40, 0, 56, 38, 24, 21, 35, 38, 0, 58, 31, 49, 38, 25, 34, 0 , 52, 40, 42, 40, 39, 54, 0, 64, 44, 50, 38, 37, 32. For the presented series, find the arithmetic mean, mode, range and median. For the presented series, find the arithmetic mean, mode, range and median.
Homework. At the figure skating competitions, the athlete's performance was assessed with the following points: At the figure skating competitions, the athlete's performance was assessed with the following points: 5.2; 5.4; 5.5; 5.4; 5.1; 5.1; 5.4; 5.5; 5.3. 5.2; 5.4; 5.5; 5.4; 5.1; 5.1; 5.4; 5.5; 5.3. For the resulting series of numbers, find the arithmetic mean, range and mode. For the resulting series of numbers, find the arithmetic mean, range and mode.
First level
Statistics. Basic concepts and definitions (2019)
Lyudmila Prokofievna Kalugina (or simply “Mymra”) in the wonderful film “Office Romance” taught Novoseltsev: “Statistics is a science, it does not tolerate approximation.” To avoid getting hit hot hand strict boss Kalugina (and at the same time easily solve tasks from the Unified State Examination 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.
So what is Statistics and why is it needed? The word "statistics" comes from Latin word“status” (status), which means “the state and state of affairs/things”. Statistics deals with the study of the quantitative side of mass social phenomena and processes in numerical form, identifying special patterns. Today statistics is used in almost all areas public life, ranging from fashion, cooking, gardening to astronomy, economics, 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:
- sample size,
- sample range,
- average,
- fashion,
- median,
- frequency,
- 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 amount ( total) of all solved problems 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 in the definition of median important point, which we haven’t met here yet? 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 following 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:
- bar chart,
- pie chart,
- bar chart,
- 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(on horizontal axis grades are reflected on the vertical axis; we plot 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.
To the exhibition innovative technologies Representatives of the companies arrived. 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
ELEMENTS OF STATISTICS. BRIEFLY ABOUT THE MAIN THINGS.
Sample size- the number of elements in the sample.
Sample range- difference between maximum and minimum values 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- number of repetitions certain value parameter in the selection.
Relative frequency
For clarity, it is convenient to present data in the form of appropriate charts/graphs
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 total number data in a row.
| Average | |
| The arithmetic mean of a series of numbers is the quotient of dividing the sum of these numbers by the number of terms. | Determine how many parts workers produced on average per shift: (23+20+25+20+23+25+35+37+34+23+30+29):12=324:12=27(min) 27 -arithmetic mean of the series under consideration. |
| Scope | |
| The range of a series of numbers is the difference between the largest and smallest of these numbers. Range = largest number – smallest smaller number | Largest quantity parts 37 Smallest – 20 parts Range = 37 – 20 = 17 parts. |
| Fashion | |
| Fashion A series of numbers is the number that appears most frequently in a given series. | 23; 20; 25; 20; 23; 25; 35; 37; 34; 23; 30; 29 A common number is 23. 23 – fashion the series under consideration. |
| Median is a number that divides a set of numbers into two parts of equal size. | Algorithm for finding the median of a set of numbers: Arrange a number set (make a ranked series). Simultaneously cross out the “largest” and “smallest” numbers of a given set of numbers until one or two numbers remain. If there is one number left, then it is the median. If there are two numbers left, then the median will be the arithmetic mean of the two remaining numbers. 23; 20; 25; 20; 23; 25; 35; 37; 34; 23; 30; 29 20; 20 ; 23 ; 23 ; 23 ; 25; 25; 29 ; 30 ; 34 ; 35; 37 The median of this series is: (25+25): 2=25. |
Arithmetic mean, range and mode, median.
Having carried out a record of parts manufactured per shift by workers of one team, we received the following series of data:
23; 20; 25; 20; 23; 25; 35; 37; 34; 23; 30; 29
Tasks for independent solution
The height (in centimeters) of five students is recorded: 158, 166, 134, 130, 132. How much does the arithmetic mean of this set of numbers differ from its median?
During the quarter, Ira received the following marks in mathematics: three “twos”, two “threes”, ten “fours” and five “fives”. Find the sum of the arithmetic mean and the median of its estimates.
The height (in centimeters) of five students is recorded: 149, 136, 163, 152, 145. Find the difference between the arithmetic mean of this set of numbers and its median?
The ages (in years) of seven employees are recorded: 25, 37, 42, 24, 33, 50, 27. How many
does the arithmetic mean of this set of numbers differ from its median?
Dollar exchange rate during the week: 30.48; 30.33; 30.45; 30.28; 30.37; 30.29; 30.34. Find the median of this series.
Every half hour, a hydrologist measures the temperature of the water in the reservoir and receives
the following series of values: 12.8; 13.1; 12.7; 13.2; 12.7; 13.3; 12.6; 12.9; 12.7; 13; 12.7. Find the median of this series.
Price meat dishes in the cafe represents a row: 198; 214; 222; 224; 229; 173; 189. Find the difference between the arithmetic mean and the median of this series.
Students of the class behind test in algebra the following grades were obtained:
3; 4; 4; 4; 2; 5; 5; 5; 3; 3; 4; 3; 3; 5; 4. Find the difference between the arithmetic mean and the median of this series.
The air temperature in Moscow during the week ranged from 23, 25, 27, 24, 21, 28, 27 degrees below zero. Find the sum of the median and range of this series of numbers.
At the shooting competition, 9th grade students showed results
representing the series 82, 49, 61, 77, 58, 42 points. Find the arithmetic mean of this series of numbers.
The sale of fruit in a store for a week represents the range 345, 229, 456, 358, 538, 649, 708 kg per day. Find the difference between the median and the arithmetic mean of this series of numbers.
The increase in prices for some products represents a series of 3.4; 6.5; 2.8; 3.7; 5.1; 4.1; 5.9 percent. Find the difference between the median and range of this series of numbers.
The transport agency recorded the number of orders for cargo delivery for 6 days. We received the following series of data: 40, 41, 39, 36, 41, 31. How different is the mode of this set of numbers from its arithmetic mean?
The bowler made 5 shots and hit 8, 9, 7, 10, 6 pins. Find the average
arithmetic of this series of numbers.
The average temperature in January is -18 degrees, in February -15 degrees, in March -7 degrees, in April +12 degrees. Find the arithmetic mean of this series of numbers.
Answers
7,85
30,34
12,8
0,2
61,5
0,4
The date of the __________
Lesson topic: Arithmetic mean, range and mode.
Lesson objectives: repeat the concepts of such statistical characteristics as the arithmetic mean, range and mode, develop the ability to find the average statistical characteristics of various series; develop logical thinking, memory and attention; to instill diligence, discipline, perseverance, and accuracy in children; develop children's interest in mathematics.
During the classes
Class organization
Repetition ( Equation and its roots)
Define an equation with one variable.
What is the root of an equation?
What does it mean to solve an equation?
Solve the equation:
6x + 5 =23 -3x 2(x - 5) + 3x =11 -2x 3x - (x - 5) =14 -2x
Updating knowledge repeat the concepts of such statistical characteristics as arithmetic mean, range, mode and median.
Statistics is a science that deals with the collection, processing, and analysis of quantitative data about a variety of mass phenomena occurring in nature and society.
Average - is the sum of all numbers divided by their number. (The arithmetic mean is called the average value of a number series.)
Range of numbers is the difference between the largest and smallest of these numbers.
Mode of number series - This is the number that appears in a given series more often than others.
Median an ordered series of numbers with an odd number of terms is called the number written in the middle, and with an even number of terms is called the arithmetic mean of the two numbers written in the middle.
The word statistics is translated from Latin language status - state, state of affairs.
Statistical characteristics: arithmetic mean, range, mode, median.
Learning new material
Task No. 1: 12 seventh grade students were asked to record the time (in minutes) spent on algebra homework. We received the following data: 23,18,25,20,25,25,32,37,34,26,34,25. On average, how many minutes did students spend on homework?
Solution: 1) find the arithmetic mean:
2) find the range of the series: 37-18=19 (min)
3) fashion 25.
Task No. 2: In the city of Schaslyve they measured daily at 18 00 air temperature (in degrees Celsius for 10 days) as a result of which the table was filled in:
T Wed = 0 WITH,
Range = 25-13=12 0 WITH,
Task No. 3: Find the range of numbers 2, 5, 8, 12, 33.
Solution: Largest number here 33, the smallest is 2. This means the range is: 33 – 2 = 31.
Task No. 4: Find the mode of the distribution series:
a) 23 25 27 23 26 29 23 28 33 23 (mode 23);
b) 14 18 22 26 30 28 26 24 22 20 (modes: 22 and 26);
c) 14 18 22 26 30 32 34 36 38 40 (no fashion).
Task No. 5 : Find the arithmetic mean, range and mode of the series of numbers 1, 7, 3, 8, 7, 12, 22, 7, 11,22,8.
Solution: 1) The number 7 appears most often in this series of numbers (3 times). It is the mode of a given series of numbers.
Solution of exercises
A) Find the arithmetic mean, median, range and mode of a series of numbers:
1) 32, 26, 18, 26, 15, 21, 26;
2) 21, 18, 5, 25, 3, 18, 5, 17, 9;
3) 67,1 68,2 67,1 70,4 68,2;
4) 0,6 0,8 0,5 0,9 1,1.
B) The arithmetic mean of a series consisting of ten numbers is 15. The number 37 was added to this series. What is the arithmetic mean of the new series of numbers?
IN) In the series of numbers 2, 7, 10, __, 18, 19, 27, one number turned out to be erased. Reconstruct it, knowing that the arithmetic mean of this series of numbers is 14.
G) Each of the 24 participants in the shooting competition fired ten shots. Noting each time the number of hits on the target, we received the following series of data: 6, 5, 5, 6, 8, 3, 7, 6, 8, 5, 4, 9, 7, 7, 9, 8, 6, 6, 5 , 6, 4, 3, 6, 5. Find the range and mode for this series. What characterizes each of these indicators?
Summarizing
What is the arithmetic mean? Fashion? Median? Scope?
Homework:
№164 (repetition task), pp. 36-39 read
№167(a,b), No. 177, 179