Notes

Outline

Part 1: Probability Distributions Probability Distribution Normal Distribution Student-t Distribution Chi Square Distribution F Distribution Significance Tests

Purposes of Data Analysis

Parameters and Statistics Parameters: Numbers that describe a population. For example, the population mean (m)and standard deviation (s).      Statistics:  Numbers that are calculated from a sample. A given population has only one value of a particular parameter, but a particular statistic calculated from different samples of the population has values that are generally different, both from each other, and from the parameter that  the statistics is designed to estimate. The science of statistics is concerned with how to draw reliable, quantifiable inferences from statistics about parameters.

Variables and Samples Random Variable: A variable whose values occur at random, following a probability distribution. Observation: When the random variable actually attains a value, that value is called an observation (of the variable). Sample: A collection of several observations is called sample. If the observations are generated in a random fashion with no bias, that sample is known as a random sample.

Probability Distribution The pattern of probabilities for a set of events is called a probability distribution. The probability of each event or combinations of events must range from 0 to 1. The sum of the probability of all possible events must be equal too 1.

Example  If you throw a die, there are six possible outcomes: the numbers 1, 2, 3, 4, 5 or 6. This is an example of a random variable (the dice value), a variable whose possible values occur at random.  When the random variable actually attains a value (such as when the dice is actually thrown) that value is called an observation.  If you throw the die 10 times, then you have a random sample which consists of 10 observations.

Probability Density Function P = the probability that a randomly selected value of a variable X falls between a and b.      f(x) = the probability density function. The probability function has to be integrated over distinct limits to obtain a probability. The probability for X to have a particular value is ZERO. Two important properties of the probability density function:      (1) f(x) ¾ 0 for all x within the domain of f.      (2)

Cumulative Distribution Function The cumulative distribution function F(x) is defined as the probability that a variable assumes a value less than x. The cumulative distribution function is often used to assist in calculating probability (will show later). The following relation between F and P is essential for probability calculation:

Normal Distribution f: probability density function      μ: mean of the population      σ: standard deviation of the population The normal distribution is one of the most important distribution in geophysics. Most geophysical variables (such as wind, temperature, pressure, etc.) are distributed normally about their means.

Standard Normal Distribution The standard normal distribution has a mean of 0 and a standard deviation of 1. This probability distribution is particularly useful as it can represent any normal distribution, whatever its mean and standard deviation. Using the following transformation, a normal distribution of variable X can be converted to the standard normal distribution of variable Z:                                          Z = ( X - μ ) / σ

Transformations It can be shown that any frequency function can be transformed in to a frequency function of given form by a suitable transformation or functional relationship. For example, the original data follows some complicated skewed distribution, we may want to transform this distribution into a known distribution (such as the normal distribution) whose theory and property are well known. Most geoscience variables are distributed normally about their mean or can be transformed in such a way that they become normally distributed. The normal distribution is, therefore, one of the most important distribution in geoscience data analysis.

How to Use Standard Normal Distribution Example 1: What is the probability that a value of Z is greater than 0.75?       Answer: P(Z³0.75) = 1-P(Z£0.75)=1-0.7734=0.2266

Another Example Example 2: What is the probability that Z lies between the limits Z1=-0.60 and Z2=0.75? Answer:     P(Z £-0.60) = 1 - P(Z < 0.60) è P(Z >-0.60)=1-0.7257=0.2743     P(-0.60 £Z £0.75) = P(Z £0.75) - P(Z £-0.60)                                   = 0.7734 - 0.2743                                   = 0.4991

Example 1

Example 2

Probability of Normal Distribution

Probability of Normal Distribution

Application in Operational Climatology This figure shows average temperatures for January 1989 over the US, expressed as quantiles of the local Normal distributions. Different values of m and s have been estimated for each location.

How to Estimate m and s In order to use the normal distribution, we need to know the mean and standard deviation of the population But they are impossible to know in most geoscience applications, because  most geoscience populations are infinite. We have to estimate m and s from samples.

Sampling Distribution Is the sample mean close to the population mean? To answer this question, we need to know the probability distribution of the sample mean. We can obtain the distribution by repeatedly drawing samples from a population and find out the frequency distribution.

Example If we repeatedly draw N observations from a standard normal distribution (m=0 and s=1) and calculate the sample mean, what is the distribution of the sample mean?

Distribution of Sample Average What did we learn from this example? èIf a sample is composed of N random observations from a normal distribution with mean m and standard deviation s, then the distribution of the sample average will be a normal distribution with mean m but standard deviation s/sqrt(N).

Standard Error The standard deviation of the probability distribution of the sample mean (X) is also referred as the “standard error” of X.

Distribution of Sample Means One example of how the normal distribution can be used for data that are “non-normally” distributed is to determine the distribution of sample means. Is the sample mean close to the population mean? To answer this question, we need to know the probability distribution of the sample mean. We can obtain the distribution by repeatedly drawing samples from a population and find out the frequency distribution.

An Example of Normal Distribution If we repeatedly draw a sample of N values from a population and calculate the mean of that sample, the distribution of sample means tends to be in normal distribution.

Central Limit Theory In the limit, as the sample size becomes large, the sum (or the mean) of a set of independent measurements will have a normal distribution, irrespective of the distribution of the raw data. If there is a population which has a mean of mand a standard deviation of s, then the distribution of its sampling means will have:  a mean m x = m, and  a standard deviation sx = s/(N)1/2 = standard error of the mean

Sample Size The bigger the sample size N, the more accurate the estimated mean and standard deviation are to the true values. How big should the sample size is for the estimate mean and standard deviation to be reasonable?       è N > 30 for s to approach s (m gets approached even with small N)

Normal Distribution of Sample Means The distribution of sample means can be transformed into the standard normal distribution with the following transformation: The confidence level of the sample mean can therefore be estimated based on the standard normal distribution and the transformation.

The Importance of the Central Limit Theorem With this theorem, statistician can make reasonable inferences about the sample mean without having to know the underlying probability distribution.

Small Sampling Theory When the sample size is smaller than about 30 (N<30), we can not use the normal distribution to describe the variability in the sample means. With small sample sizes, we cannot assume that the sample-estimated standard deviation (s) is a good approximation to the true standard deviation of the population (s). The quantity            no longer follows the standard normal distribution. In stead, the probability distribution of the small sample means follows the “Student’s t distribution”.

Student’s t Distribution For variable t The probability density function for the t-distribution is: Here K(n) is a constant which depends on the number of degree of freedom (n = n-1). K(n) is chosen so that:

How to Use t-Distribution

Slide 33

Student’s t and Normal Distributions Since the Student’s t distribution approaches the normal distribution for large N, there is no reason to use the   normal distribution in preference to Student’s t. The Student ’s t distribution is the most commonly used in meteorology, and perhaps in all of applied statistics.

Slide 35

How to Determine DF? The number of degree of freedom      = (the number of original observations) – (the number of parameters estimated from the observation) For example, the number of degree of freedom for a variance is N-1, where N is the number of the observations. It is because we need to estimate the mean (a parameter) from the observations, in order to calculate the variance.

Example We have four observations: [4, 1, 8, 3] To calculate the variance, we first get the mean = (4+1+8+3)/4=4 We then calculate the deviations of each observations from the mean and get d = [4-4, 1-4, 8-4, 3-4] = [0, -3, 4, -1]. Now these four d’s are constrained by the requirement:       d1 + d2 + d3 + d4 =0 (a result related to the mean). The variance we get (variance = d1**2+d2**2+d3**2+d4**2) has only 3 degree of freedom ( 4 observations minus one estimate of the parameter).

Statistical Inference Two of the main tools of statistical inference are: Confidence Intervals     Within what intervals (or limits) does X% of the population lie in the  distribution Hypothesis Tests     You formulate a theory (hypothesis) about the phenomenon you are studying and examine whether the theory is supported by the statistical evidence.

Confidence Intervals (or Limits) In stead of asking “what is the probability that Z falls within limits a and b in the normal distribution”, it is more important to ask “Within what intervals or limits does X% of the population lie in the normal distribution”. The X% is referred as the “confidence level”. The interval correspond to this level is called the “confidence intervals” or “confidence limits”.

Normal Distribution of Sample Means The distribution of sample means can be transformed into the standard normal distribution with the following transformation: The confidence level of the sample mean can therefore be estimated based on the standard normal distribution and the transformation.

How to Determine Confidence Intervals Choose a confidence level Determine Z-values from the table of standard normal distribution. (Note that Z1 = - Z2) Transform Z values back to X values using Z = (X - m ) / s The Confidence intervals are determined.

Confidence Intervals for Sample Mean The probability that a sample mean, x, lies between m-sx and m+ sx may be written: However, it is more important to find out the confidence interval for the population mean m.. The previous relation can be written as: But we don’t know the true value of s (the standard deviation of the population). If the sample size is large enough (N³30), we can use s (the standard deviation estimated by samples) to approximate s:

Confidence Intervals for Sample Mean The 68% confidence intervals for m are: To generalize the relation to any confidence level X%, the previous relation can be rewritten as:

An Example In an experiment, forty measurements of air temperature were made. The mean and standard deviation of the sample are:                               x = 18.41°C    and   s = 0.6283°C. Question 1:      Calculate the 95% confidence interval for the population mean for these data. Answer:      From the previous slid, we know the interval is: x ± Z95% * s / (N)0.5      Z95% = 1.96      Z95% * s / (N)0.5 = 1.96 * 0.6283°C / (40 )0.5 = 0.1947°C      The 95% confidence level for the population mean: 18.22°C ~ 18.60°C.

An Example – cont. In an experiment, forty measurements of air temperature were made. The mean and standard deviation of the sample are:                               x = 18.41°C    and   s = 0.6283°C. Question 2:      How many measurements would be required to reduce the 95% confidence interval for the population mean to (x-0.1)°C to (x+0.1)°C? Answer:       We want to have Z95% * s / (N)0.5 = 0.1 °C       We already know Z95% = 1.96 and s = 0.6283°C       è N = (1.96´0.6283/0.1)2 = 152

Application of Confidence Intervals A typical use of confidence intervals is to construct error bars around plotted sample statistics in a graphical display.

Interpreting the Confidence Intervals The term 95% confident means that we are confident our procedure will capture the value of m 95% of the times it (the procedure                            ) is used.

Hypothesis (Significance) Testing Hypothesis testing involves comparing a hypothesized population parameter with the corresponding number (or statistic) determined from sample data. The hypothesis testing is used to construct confidence interval around sample statistics. The above table lists the population parameters that are often tested between the population and samples.

Parametric .vs. Nonparametric Tests Parametric Tests: conducted in the situations where one knows or assumes that a particular theoretical distribution (e.g., Normal distribution) is an appropriate representation for the data and/or the test statistics. è In these tests, inferences are made about the particular distribution parameters. Nonparametric Test: conducted without the necessity of assumption about what theoretical distribution pertains to the data.

Significance Tests Five Basic Steps:     1. State the null hypothesis and its alternative     2. State the statistics used     3. State the significance level     4. State the critical region         (i.e., identify the sample distribution of the test statistic)     5. Evaluate the statistics and state the conclusion

Null Hypothesis Usually, the null hypothesis and its alternative are mutually exclusive. For example:      H0: The means of two samples are equal.      H1: The means of two samples are not equal.      H0: The variance at a period of 5 days is less than or equal to C.      H1: The variance at a period of 5 days is greater than C.

Examples

Rejection/Acceptance of Hypothesis

Test Statistics A test statistic is the quantity computed from the sample that will be the subject of the test. In parametric tests, the test statistic will often be the sample estimate of a parameter of a relevant theoretical distribution.

Example To examine the significance of the difference between means obtained from two different samples. We use the  “two-sample t-statistic”:

Confidence and Significance Confidence Level: X% Significance Level: a

What Does It Mean 5% Significant? At the 5% significance level, there is a one in twenty chance of rejecting the hypothesis wrongly (i.e., you reject the hypothesis but it is true).

Sampling Distribution A statistic is calculated from a batch of data. The value of the statistic varies from batch to batch. A sampling distribution for a statistic is the probability distribution describing batch-to-batch variations of that statistic. The random variations of sample statistics can be described using probability distributions just as the random variations of the underlying data can be described using probability distributions. Sample statistics can be viewed as having been drawn from probability distributions.

Critical Regions

p Value The p value is the specific probability that the observed value of the test statistic will occur according to the null distribution (the probability distribution based on the null hypothesis).  If p value falls within the critical regions      è Reject the null hypothesis. The p value depends on the alternative hypothesis.

Example Ho: m = 50       H1: m ¹ 50 If a N=25 sampling mean has a sample mean of 45 and a sample s of 15, then         Z = (45-50) / (15/5) = -1.67 P(Z<-1.67)=0.0478 p value = 2 * 0.0478 = 0.0956 Ho: m = 50       H1: m < 50         Z = (45-50) / (15/5) = -1.67 P(Z<-1.67)=0.0478 p value = 0.0478 = 0.0478

Rejection/Acceptance of Hypothesis

Type-I and Type-II Errors Type I Error: You reject a null hypothesis that is true. Type II Error: You fail to reject a null hypothesis that is false. By choosing a significance level of, say, a=0.05, we limit the probability of a type I error to 0.05.

One-Side and Two-Side Tests Two-Side Tests       H0: mean temperature = 20°C       H1: mean temperature ¹ 20°C

Critical Regions

Example

Significance Test of the Difference of Means We want to compare the averages from two independent samples to determine whether a significance exists between the samples. For Example:      * One sample contains the cholesterol data on patients taking a standard drug, while the second sample contains cholesterol data on patients taking experimental drug. You would test to see whether there is statistically significant difference between two sample averages.      * Compare the average July temperature at a location produced in a climate model under a doubling versus no doubling CO2 concentration.      * Compare the average winter 500-mb height when one or the other of two synoptic weather regimes had prevailed.

Two kinds of Two-Sample t-Test For samples come from distributions with different standard deviations, having values of s1 and s2. This is called “unpooled two-sample t-statistic”: If both distributions have the same standard deviation, then we can “pool” the estimates of the standard deviation from the two samples into a single estimate:

Paired Data (One-Sample) t-Test Paired Data: where observations come in natural pairs.      For example:              A doctor might measure the effect of a drug by measuring              the physiological state of patients before and after applying              the drug. Each patient in this study has two observations, and              the observations are paired with each other. To determine the              drug’s effectiveness, the doctor looks at the difference               between the before and the after reading. Paired-data test is a one-sample t-test      To test whether or not there is a significant difference, we use      one-sample t-test. It is because we are essentially looking at      one sample of data – the sample of paired difference.

Example There is a data set that contains the percentage of women in the work force in 1968 and 1972 from a sample of 19 cities. There are two observations from each city, and the observations constitute paired data. You are asked to determine whether this sample demonstrates a statistically significant increase in the percentage of women in the work force. Answer:     1. Ho: m = 0 (There is no change in the percentage)         H1: m¹ 0 (There is some change, but we are not assuming the                         direction of change)     2. 95% t-Test with N=19 è We can determine t0.25% and t0.975%     3. S=0.05974 è s = S/(19)0.5 = 0.01371     4. t statistic t = (0.0337 – 0) / s = 2.458 > t0.975%       5. We reject the null hypothesis è There has been a significant change          in women’s participation in the work force in those 4 years.

Slide 71

Test of Sampling Variance Standard deviation is another important population parameter to be tested. The sample variances have a Chi-Squared (c2) distribution, with the following definition:   Here N: sample size                  S: the sampled standard deviation                  s: the true  standard deviation

c2  Distribution

Example

Significance Test of the Difference of Variances We want to know if the standard deviation estimated from two samples are significantly different. The statistical test used to compare the variability of two samples is the F-test:      S12: the estimate of the population variance of sample 1      S22: the estimate of the population variance of sample 2.

F-Distribution Here:       X  =  S12/S22          n1 = degree of freedom of sample 1          n2 = degree of freedom of sample 2        K(n1 , n2 ) = a constant to make the                              total area under the                              f(x) curve to be 1

F- -Distribution F-distribution is non-symmetric. There are two critical F values of unequal magnitude. Whether to use the upper or the lower critical value depends on the relative magnitudes of S1 and S2:      If S1 > S2, then use the upper limit FU to reject the null hypothesis      If S1 < S2, then use the lower limit FL to reject the null hypothesis Or you can always make S1 > S2 when F is defined. In this case, you can always use the upper limit to examine the hypothesis.

Slide 78

Example There are two samples, each has six measurements of wind speed. The first sample measures a wind speed variance of 10 (m/sec)2. The second sample measures a 6 (m/sec)2 variance. Are these two variance significantly different? Answer: (1) Selected a 95% significance level (a = 0.05)                (2) H0: s1 = s2                      H1: s1 ¹ s2                (3) Use F-test                      F = 10/6 = 1.67                (4) This is a two-tailed test. So choose the 0.025 significance areas.                      For n1 = n1 = 6-1=5, F0.975=7.15.                (5) Since F< F0.975, the null hypothesis can not be rejected.                (6) Variance from these two samples are not significantly different.