Knowledge points of high school mathematics statistics: statistics 1. 1. 1 simple random sampling.
1. Population and sample
In statistics, the whole research object is called population.
Call each research object an individual.
The total number of individuals in a group is called the total capacity.
In order to study the related properties of the population x, a part of the population is generally randomly selected: x? X, xn study, we call it sample. The number of individuals is called sample size.
2. Simple random sampling, also called pure random sampling. I.e. without any grouping, sorting, queuing, etc. In general, the investigation unit is completely selected by the machine. The characteristics are: the probability of each sample unit being extracted is the same (the probability is equal), and each unit of the sample is completely independent, and there is no certain correlation and exclusion between them. Simple random sampling is the basis of other sampling forms. This method is usually only used when the difference between the whole units is small and the number is small.
3. Common methods of simple random sampling:
(1) draw lots; (2) Random number table method; ⑶ Computer simulation method; ⑷ Direct extraction with statistical software. In the sample size design of simple random sampling, the main considerations are: ① population variation; ② Allowable error range; ③ Degree of probability assurance.
4. Draw lots:
(1) Number each object in the investigation team.
(2) Prepare the lottery tool and implement the lottery.
(3) Measure or investigate each individual in the sample.
Please investigate the favorite sports activities of students in your school.
5. Random number table method:
Example: Select 10 students from the class to participate in an activity by using a random number table.
1. 1.2 system sampling
1. System sampling (equidistant sampling or mechanical sampling):
Sort the units of the population, then calculate the sampling distance, and then sample according to this fixed sampling distance. The first sample was selected by simple random sampling.
K (sampling distance) =N (overall size) /n (sample size)
Prerequisite: For the variables studied, the arrangement of individuals in the group should be random, that is, there is no regular distribution related to the variables studied. You can start sampling from different samples and compare the characteristics of several samples under the conditions allowed by the investigation. If there are obvious differences, it shows that the distribution of samples in the population follows a certain cycle law, and this cycle coincides with the sampling distance.
2. Systematic sampling, namely equidistant sampling, is one of the most commonly used sampling methods in practice. Because it has low requirements for sampling frames and simple implementation. More importantly, if there are some auxiliary variables related to the survey indicators available, and the whole unit is queued according to the size of the auxiliary variables, systematic sampling can greatly improve the estimation accuracy.
1. 1.3 stratified sampling
1. stratified sampling (type sampling):
First of all, according to some characteristics or signs (gender, age, etc.), all units in the group are divided into several types or levels. ), and then extract a sub-sample from each type or level by simple random sampling or systematic sampling. Finally, these sub-samples are combined to form a total sample.
Two methods:
1. Firstly, the population is divided into several layers by stratification variables, and then extracted from each layer according to the proportion of each layer in the population.
2. Firstly, the population is divided into several layers by stratification variables, and then the elements in each layer are arranged neatly in hierarchical order. Finally, samples are extracted by systematic sampling.
2. Stratified sampling is to divide people with strong heterogeneity into sub-populations with strong homogeneity, and then draw samples from different sub-populations to represent sub-populations, and all samples represent people again.
Stratification standard:
(1) Take the main variables or related variables to be analyzed and studied in the investigation as the standard of stratification.
(2) Ensure that the variables with strong homogeneity in each layer, strong interlayer heterogeneity and outstanding overall internal structure are used as stratified variables.
(3) Take those variables with obvious stratification as hierarchical variables.
3. The proportion of stratification:
(1) Proportional stratified sampling: a method to extract sub-samples according to the proportion of units of various types or levels to the total units.
(2) Non-proportional stratified sampling: If the proportion of some levels in the population is too small, the sample size will be very small. At this time, this method is mainly used to facilitate special research or comparison of different levels of subpopulations. If we want to infer the population from the sample data, we need to first weight the data of each layer, adjust the proportion of each layer in the sample, and restore the data to the actual proportion structure of each layer in the population.
Knowledge points of high school mathematics statistics: probability 2. 1. 1? 2. 1.2 Probability of random events and its significance
1, basic concept:
(1) inevitable event: the event that will happen under condition S is called the inevitable event relative to condition S;
(2) Impossible events: events that will not happen under condition S are called impossible events relative to condition S;
(3) Deterministic events: inevitable events and impossible events are collectively referred to as deterministic events relative to condition S; (4) Random events: events that may or may not occur under condition S are called random events relative to condition S;
(5) Frequency and frequency: repeat the test for n times under the same condition S, and observe whether there is an event A, and call the frequency nA of the event A in the n tests as the frequency of the event A; Call the occurrence ratio of event A the occurrence probability of event A: for a given random event A, if the occurrence frequency fn(A) of event A is stable at a certain constant with the increase of test times, write this constant as P(A) and call it the probability of event A. ..
(6) Difference and connection between frequency and probability: The frequency of a random event refers to the ratio of the number of times nA of the event to the total number of times n of testing, which has certain stability and always swings around a certain constant, and with the increase of testing times, the swing amplitude becomes smaller and smaller. We call this constant the probability of random events, which quantitatively reflects the probability of random events. Frequency can be approximated as the probability of the event under the premise of a large number of repeated experiments.
2. Basic Properties of1.3 Probability
1, basic concept:
The inclusion, union, intersection and equality of (1) events.
(2) if a? B is an impossible event, that is, a? B =ф Ф, then event A and event B are mutually exclusive;
(3) if a? B is an impossible event, a? B is an inevitable event, then event A and event B are mutually opposite events;
(4) When events A and B are mutually exclusive, the addition formula is satisfied: P(A? B)= P(A)+P(B)； If events A and B are opposite events, then A? B is an inevitable event, so P(A? B)= P(A)+ P(B)= 1, so there is P(A)= 1? P(B)
2, the basic nature of probability:
1) The probability of an inevitable event is 1, and the probability of an impossible event is 0, so 0? P(A)？ 1;
2) When events A and B are mutually exclusive, the addition formula satisfies: P(A? B)= P(A)+P(B)；
3) If events A and B are opposite events, then A? B is an inevitable event, so P(A? B)= P(A)+ P(B)= 1, so there is P(A)= 1? p(B)；
4) The difference and connection between mutually exclusive events and opposing events, mutually exclusive events means that in an experiment, event A and event B will not happen at the same time, including three different situations: (1) Event A happens and event B doesn't happen; (2) Event A does not occur, but Event B does; (3) Event A and Event B do not occur at the same time, but the opposite event means that there is only one event A and Event B, including two situations: (1) Event A occurs and Event B does not; (2) Event B happens and Event A doesn't, which is a special case of mutually exclusive events.
High school mathematical statistics knowledge point 1, scientific notation: write a number notation in form.
2. Statistical chart: a chart that visually represents the collected data.
3. Sector statistical chart: circle and sector are used to represent the relationship between the whole and the part, and the size of the sector reflects the percentage of the part in the whole; In a fan-shaped statistical chart, the percentage of each part in the total is equal to the central angle of the fan-shaped corresponding to that part and 360? The proportion of.
4. Bar chart: clearly show the specific figures of each item.
5, broken line statistical chart: clearly reflect the changes of things.
6. Some events include: inevitable events that will definitely happen and impossible events that will definitely not happen.
7. Uncertain events: events that may or may not occur; The possibility of uncertain events is different; Not sure.
8. Event probability: Theoretical probability can be obtained by dividing the event result by all possible results.
9. Significant digits: for approximation, from the first digit on the left that is not 0 to the nearest digit.
10, both sides of the game are fair: both sides have the same possibility of winning.
1 1, arithmetic mean: abbreviation? General? , the most commonly used, greatly influenced by extreme values; Weighted average 12, median: the data are arranged by size, and the number in the middle position is simple to calculate, which is less affected by extreme values.
13, mode: the data with the highest frequency in a group of data is less affected by extreme values and has little to do with other data.
14, mean, mode and median all represent data and describe a set of data. General? .
15, census: conduct a comprehensive census of the respondents for a certain purpose; All subjects are called whole, and each subject is called individual.
16, sampling survey: select some individuals from the population for investigation; Some individuals extracted from the population are called samples (representatives).
17. random survey: according to the principle of equal opportunity, the probability of each individual being investigated is the same.
18, frequency: the number of times the object appears each time.
19. Frequency: the ratio of the number of times an object appears to the total number of times.
20. Grade difference: the difference between the largest data and the smallest data in a group of data, which describes the degree of data dispersion.
2 1, variance: the average of the square of the difference between each data and the average, which describes the degree of dispersion of the data.
22, variance calculation formula
23. Standard Variance: The arithmetic square root of variance describes the dispersion degree of data.
24. The smaller the grade difference, variance and standard deviation of a set of data, the more stable this set of data will be.
25. Find out the probability of an event with a tree diagram or table.