獨(dú)家Alevel數(shù)學(xué)Statistic1S1考試知識(shí)點(diǎn)

1、Summary of key points in S1 Chapter 1: Binomial distribution1. （重點(diǎn)*）計(jì)算二項(xiàng)分布的概率： （1）公式法（*），由，則有 （2）查表法（*）：利用書(shū)中135-139頁(yè)中的，其中p 是0.05的倍數(shù)、一直到0.50，n最小是5、最大是50。2. （重點(diǎn)*）計(jì)算二項(xiàng)分布的期望和方差：，則有 3. （考點(diǎn)*）二項(xiàng)分布的條件：l A fixed number of trials,.l Each trial should be success or failure.l The trials are independent.l The pr

2、obability of success, at each trial is constant. 其中，為指數(shù)（index），為參數(shù)（parameter）。難點(diǎn)是要求根據(jù)題意寫(xiě)出二項(xiàng)分布的條件，如果有題意背景的，要根據(jù)題意寫(xiě)。4. （考點(diǎn)*）如果，其中，則，那么；如果p是0.05的倍數(shù)，則可以用查表法求概率。5. 典型例題：例7/8/9*/10/11/12/13(a)/14*6. 復(fù)習(xí)題：Review Exercise 1: 1/4/87. 練習(xí)冊(cè)部分題目： 12-01-2, 10-01-1, 08-01-2Chapter 2: Representation and summary of da

3、ta location1、Frequency tables and grouped datacumulative frequency：to add a column to the frequency table showing the running total of the frequencies.A grouped frequency distribution consists of classes and their related class frequencies.Classes 30-31 32-33 34-35For the class 32-33Lower class boun

4、dary is 31.5Upper class boundary is 33.5Class width is 33.5-31.5=2Class mid-point is (31.5+33.5)/2=32.52、The measurements of location of the centre of a set of data mode, median and meanl The mode is the value that occurs most often.l The median is the middle value or the half of the two middle valu

5、es, when the data is put in order.l The mean is the sum of all the observations divided by the total number of the observations.The mean of a sample of data in a frequency distribution, is where 3、Coding for large data valuesCoding is normally of the form where and are to be chosen.To find the mean

6、of the original data; find the mean of the coded data, equate this to the coding used and solve. Chapter 3:Representation and summary of data measures if dispersion1、The range of a set of data is the difference between the highest and lowest value in the set.The quartiles, split the data into four p

7、arts. To calculate the lower quartile, divide by 4.For discrete data for the lower quartile, divide by 4. To calculate the upper quartile, divide by 4 and multiply by 3. When the result is a whole number find the mid-point of the corresponding term and the term above. When the result is not a whole

8、number round the number up and pick the corresponding term.For continuous grouped data for divide by 4, fordivide by 4 and multiply by 3. Use interpolation to find the value of the corresponding term.The inter-quartile range is 2、The standard deviation and variance of discrete datavariance=standard

9、deviation= If you let stand for the frequency, then and Variance=3、Adding or subtracting numbers does not change the standard deviation of the data.Multiplying or dividing the data by a number does affect the standard deviation.To find the standard deviation of the original data, find the standard d

10、eviation of the coded data and either multiply this by what you divide the data by, or divide this by what you multiplied the data by. Chapter 4: Representation of data1. A stem and leaf diagram is used to order and present data given to two or three significant figures. Each number is first split i

11、nto its stem and leaf.Two set of data can be compared by using back-to-back stem and leaf diagrams.2、An outlier is an extreme value that lies outside the overall pattern of the data, which is n greater that the upper quartile +1.5inter-quartile rangeorn less that the lower quartile 1.5inter-quartile

12、 range.3、Box plotUsing box plots to compare two sets of data4、HistogramA histogram gives a good picture of how data are distributed. It enables you to see a rough location, the general shape of the data and how spread out the data are.A histogram is similar to a bar chart but are two major differenc

13、esl There are no gaps between the bars.l The area of the bar is proportional to the frequency. To calculate the height of each bar (the frequency density) use the formula Area of bar=frequency. is the easiest value to use when drawing a histogram then Frequency density= 5、The shape (skewness) of a d

14、ata setThe ways of describing whether a distribution is skewed:n You can use the quartiles. If then the distribution is symmetrical. If then the distribution is positively skewed. If then the distribution is negatively skewed.n You can use the measures of location mode=median=mean describes a distri

15、bution which is symmetrical. modemedianmedianmean describes a distribution with negative skew. 6、Comparing the distributions of data setsl The IQR is often used together with the median when the data are skewed.l The mean and standard deviation are generally used when the data are fairly symmetrical

16、.Chapter 5: Probability1、Vocabulary used in probabilityA sample space is the set of all possible outcomes of an experiment.The probability of an event is the chance that the event will occur as a result of an experiment. Where outcomes are equally likely the probability of an event is the number of

17、outcomes in the event divided by the total number of possible outcomes in the sample space.2、Venn diagramsYou can use Venn diagrams to solve probability problems for two or three events. A rectangle represents the sample space and it contains closed curves that represent events.3、Using formulae to s

18、olve problemsAddition Rule Conditional probabilityThe probability of given , written , is called the conditional probability of given and so: Multiplication Rule 4、Tree diagrams5、Mutually exclusive and independent eventsWhenandare mutually exclusive, then , so The Addition Rule applied to mutually e

19、xclusive events: Ifandare independent, then: Chapter 6: Correlation6.1 Scatter diagramsIf both variables increase together they are said to be positively correlated. For a positive correlation the points on the scatter diagram increase as you go from left to right. Most points lie in the first and t

20、hird quadrants.If one variable increases as the other decreases they are said to be negatively correlated. For a negative correlation the points on the scatter diagram decrease as you go from left to right. Most points lie in the second and fourth quadrants.If no straight line (linear) pattern can b

21、e seen there is said to be no correlation. For no correlation the points on the scatter diagram lie fairly evenly in all four quadrants.Examples: 1/2/36.2 You can calculate measures for the variability of bivariate data注：上面的公式在公式本中有。Examples: 5*Lesson Three6.3 Product moment correlation coefficient

22、Examples: 6*Exercise 6B: Q4/5 6.4 Usingto determine the strength of the linear relationship between the variablesThe value of varies between and 1.If there is a perfect positive linear correlation between the two variables (all points fit a straight line with positive gradient).If there is a perfect

23、 negative linear correlation between the two variables (all points fit a straight line with negative gradient).If is zero (or close to zero) there is no linear correlation; this does not, however, exclude any other sort of relationship.Values of between 1 and 0 indicate a greater or lesser degree of

24、 positive correlation. The closer to 1 the better the correlation, the closer to 0 the worse the correlation. Values of between 1 and 0 indicate a greater or lesser degree of negative correlation. The closer to 1 the better the correlation, the closer to 0 the worse the correlation.Examples: 7Lesson

25、 Four6.5 The limitation of Examples: 8/96.6 Using coding to simplify the calculation of You can rewrite the variables and by using the coding and where and are suitable numbers to be chosen. is not affected by coding. Examples: 10*Exercise 6E: Q7/10 Exercises and homework: Review Exercise 2 Q1/4 Q5(

26、a)(b) (Jan 2012)Lecture 5Chapter 7: RegressionLesson One7.1 The rule is the equation of a straight line.If then (sometimes called the intercept) is where the line cuts the -axis and is the amount by which increases for an increase of 1 in , ( is called the gradient of the line).Examples: 1/27.2 Inde

27、pendent and dependent variablesAn independent (or explanatory) variable is one that is set independently of the the other variable. It is plotted along the -axis.An dependent (or response) variable is one whose values are determined by the values of the independent variable. It is plotted along the

28、-axis.Examples: 3Lesson Two7.3 The values of and for minimum sum of residualsFor each point on a scatter diagram you can express in terms of as where is the vertical distance from the line of best fit, is called residual.The line that minimizes the sum of the squares of the residuals is called the l

29、east squares regression line. The line is called the regression line of on .The equation of the regression line of on is: where and 注：上面的公式在公式本中有。Examples: 4*7.4 Coding is sometimes used to simplify calculationsExamples: 5*Lesson Three7.5 Applying and interpreting the regression equationA regression

30、 line can be used to estimate the value of the dependent variable for any value of the independent variable.Interpolation is when you estimate the value of a dependent variable within the range of the data.Extrapolation is when you estimate the value outside the range of the data. Values estimated b

31、y extrapolation can be unreliable.You should not, in general, extrapolate and you must view any extrapolated values with caution.Examples: 6/7*/8*Exercises and homework: Page147 Q6 Review Exercise 2 Q5/9/12/15/16/ Q5(c)-(f)(Jan 2012)Lesson FourChapter 8: Discrete random variables8.1 A variable is re

32、presented by a symbol, and it can take on any of a specified set of values.When the value of a variable is the outcome of an experiment, the variable is called a random variable.Another name for the list of all possible outcomes of an experiment is the sample space.For a random variable n is a parti

33、cular value of n refers to the probability that is equal to a particular value of .A continuous random variable is one where the outcome can be any value on a continuous scale.A discrete random variable takes only values on a discrete scale.Examples: 1/28.2 n To specify a discrete random variable co

34、mpletely, you need to know its set of possible values and the probability with which it takes each one.n You can draw up a table to show the probability of each outcome of an experiment. This is called a probability distribution.n You can also specify a discrete random variable as a function, which

35、is known as a probability function.Examples: 38.3 Sum of probabilitiesn For a discrete random variable the sum of all the probabilities must add up to one, that is Examples: 4Lecture 6Chapter 8: Discrete random variables8.4-8.11 Lesson One8.4Examples: 68.5 Cumulative distribution function for a disc

36、rete random variablenn Like a probability distribution, a cumulative distribution function can be written as a table.Examples: 7/8Lesson Two8.6 The mean or expected value of a discrete random variablen expected value of Examples: 9/10*8.7 Finding an expected value for n expected value of n In genera

37、l,Examples: 11*Lesson Three8.8 Finding the variance of a random variablen The variance of is usually written as Var() and is found by using: VarExamples: 12*Exercise8D: Q1/2/78.9* The expected value and variance of a function of lwhere and are constants.Examples: 13*/14*/15/16*Exercise 8E: Q1/2/3Les

38、son Four8.10 Examples: 17*Exercise8E: Q88.11For a discrete random variable over the values 1,2,3,n Examples: 18Exercises and homework: Page173 Q4/7/10 Review Exercise 2 Q2/6/8/13/18 Q3(Jan 2012)Lecture 7Chapter 9: The normal distributionLesson One9.1 Use tables to find the probability of the standar

39、d normal distributionn The standard normal distribution is written as For ,nnExamples: 1Exercise 9A: Q1/49.2 Use tables to find the value of given a probabilityn The table of percentage points of the normal distribution gives the value of for various values of n If is greater than 0.5, then If is le

40、ss than 0.5, then n If is less than 0.5, then If is greater than 0.5, then Examples: 2Lesson TwoExercise 9B: Q19.3 Transform any normal distribution into and use tablesn The normal distribution can be transformed into by the formula n You can round your value to 2 s.f. to use the nearest value in th

41、e tables.Examples: 3/4Exercise 9C: Q1/2/3Lesson Three9.4 Use normal tables to find and n If and , where is a probability, then Examples: 5/6/7Exercise9D: Q1/5Lesson Four9.5 Use the normal distribution to answer questions in contextExamples: 8*/9*Exercises and homework: Page191 Q4/6 Review Exercise 2

42、 Q3/7/11/14/17 Q7(Jan 2012)Chapter 2: Poisson distribution 1. （重點(diǎn)*）計(jì)算Poisson分布的概率： （1）公式法（*），由，則有 （2）查表法（*）：利用書(shū)中140頁(yè)中的，其中參數(shù) 最小為0.5，最大為10。2. （重點(diǎn)*）Poisson分布的條件：Events must occurl singly in space or timel independently of each otherl at a constant rate in the sense that the mean number of occurences in

43、 the interval is proportional to the length of the interval. 其中，為參數(shù)（parameter）。3. （重點(diǎn)*）用Poisson分布對(duì)二項(xiàng)分布估值的條件：如果，其中 is large and is small那么，其中4. （考點(diǎn)*）Poisson分布的期望和方差都是參數(shù)，利用比較期望和方差來(lái)判斷 是否適合用Poisson分布。5. 典型例題：例2/4/6*/7/8/10*6. 復(fù)習(xí)題：Review Exercise 1: 2/5/6/9/12/137. 練習(xí)冊(cè)部分題目： 11-01-1, 10-06-2, 10-01-3, 09-

44、06-8, 09-06-1, 08-06-5Chapter 3: Continuous random variables1. （重點(diǎn)*）概率密度函數(shù)（p.d.f.）由計(jì)算其中的未知常數(shù)；利用f(x)的定義，計(jì)算概率。2. （重點(diǎn)*）利用f(x)計(jì)算期望和方差： 3. （重點(diǎn)*）累積分布函數(shù)（c.d.f.）： （1）由f(x)求F(x)： （方法一）定積分法：對(duì)于分段函數(shù)，采用定積分的方法，見(jiàn)例5。 每段都要加上F(x)在這段上的初始值。 （方法二）不定積分法：對(duì)于分段函數(shù)，采用不定積分方法，見(jiàn)例5。 對(duì)于未知常數(shù)C，利用F(x)的特定值，尤其是0，1值點(diǎn)。 （2）由F(x)求f(x)：f(x)

45、=F(x)4. （重點(diǎn)*）利用F(x)，求中位數(shù)m、上四分位數(shù)Q1和上四分位數(shù)Q3，其中；其中，要對(duì)F(x)在節(jié)點(diǎn)處（分段函數(shù)的端點(diǎn)出）的函數(shù)值分析，采用相應(yīng)的函數(shù)表達(dá)式，同時(shí)要注意自變量的取值范圍。5. 典型例題：例2/3/5*/6*/7*/8*/10*/11/12*6. 復(fù)習(xí)題：Review Exercise 1: 3/7/10/14/167. 練習(xí)冊(cè)部分題目： 12-01-6, 11-06-7, 11-01-5/7, 10-06-4/7, 10-01-2/4, 09-06-6/7, 09-01-4/7, 08-06-7, 08-01-4/8Chapter 4: Continuous un

46、iform distribution1. （重點(diǎn)*）連續(xù)一致分布Ua,b的概率密度函數(shù)（p.d.f.） 及其函數(shù)圖像。要求由概率密度函數(shù)或者圖像判斷是否為連續(xù)一致分布。同 時(shí)，由概率密度函數(shù)求相應(yīng)的概率。2. （重點(diǎn)*）連續(xù)一致分布的期望和方差： 典型題目：（1）利用上面的公式，計(jì)算期望和方差； （2）給定期望和方差的情況下、求相應(yīng)的a、b值； （3）由公式法算得期望和方差，再由 E(X2)=Var(X)+E2(X) 求得E(X2)； （4）利用求積分的方法（往往是題目要求的）求方差和E(X2)； （5）應(yīng)用題中，利用公式法或者求積分的方法求E(X2)，并根據(jù) 題意計(jì)算與E(X2)相關(guān)的量。3

47、. （重點(diǎn)*）累積分布函數(shù)（c.d.f.）： 注意與f(x)的區(qū)分，并由它們可以判斷出是否為連續(xù)一致分布。4. 典型例題：例2/4/5/6/7/95. 復(fù)習(xí)題：Review Exercise 2: 1/4/12/206. 練習(xí)冊(cè)部分題目： 12-01-1(8), 11-06-4(4), 11-01-3(6), 10-06-3(5), 10-01-2(10), 09-01-2(9), 08-05-1(10)Chapter 5: Normal approximations1. （重點(diǎn)*）連續(xù)校正（continuity correction）：當(dāng)我們用正態(tài)分布（或者任何連續(xù)分布）對(duì)離散分布估值的時(shí)候

48、，一定要用連續(xù)校正，其主要想法就是把一個(gè)自然數(shù)看做一個(gè)小區(qū)間，分四種情況：l P(Xn)P(Yn-0.5)l P(Xn)P(Yn+0.5)l P(Xn)P(Yn+0.5)l P(Xn)P(Y10時(shí)，選擇正態(tài)分布估值。5. 典型例題：例2/4*/5/6*/76. 復(fù)習(xí)題：Review Exercise 2: 5/10/16/18/197. 練習(xí)冊(cè)部分題目： 12-01-4(e)(7), 11-06-5(e)(6)/6(b)(8), 11-01-6(e)(6), 10-06-5(c)(9), 10-01-5(b)(6), 09-06-5(b)(7), 09-01-6(c)(6), 08-06-2(

49、7)Chapter 6: Populations and samples1. （重點(diǎn)*）概念及定義題：l A population is a collection of individual items.l A census is the investigation when every member of a population is used.l A sample is a selection of individual members or items from a population.l A simple random sample, of size, is one taken so that every possible sample oof size has an equal chance of being selected.l A sampling unit is an individual member of a population.l A sampling frame is a list of sampling units used in practice to represent a population.l A statistic is a quantity calculated solely from the observ