A frequency distribution is a listing of frequencies of all categories of the observed values of a variable. When constructing a frequency distribution, as the number of classes is decreased, the class width increases. On counting all of the frequencies we obtain the desired class intervals and their frequencies. These methods are especially useful in event history analysis because patterns of variation over time and across cases are complex and are not easily captured by a single statistic. A consequence of the fact that the radon concentrations are approximately lognormally distributed is that the geometric mean of the radon concentrations is a more meaningful distribution summary than the arithmetic mean. For example, a single extra-large radon concentration found in one home may have a drastic effect on the arithmetic mean, but it will have little effect on the geometric mean.

• For larger samples, there is an approximation that is useful both in practice and in deriving methods of statistical inference.
• A consequence of the fact that the radon concentrations are approximately lognormally distributed is that the geometric mean of the radon concentrations is a more meaningful distribution summary than the arithmetic mean.
• For example, it is often found that, while the concentrations of an air or water pollutant (such as sulfur dioxide in the air or phenol in river water samples) are not nomially distributed, their logarithms are.
• Therefore, it is customary to define categories as intervals of values, which are called class intervals.

Then the probability of y successes and (n−y) failures is py(1−p)n−y. A pointwise confidence interval is the confidence interval at a single time t. Usually a simultaneous confidence band around S(t) in constructing a frequency distribution as the number of classes are decreased the class width for a time interval is preferable. The pointwise confidence intervals of ŜKM(t) and ŜKM(u) at times t and u, respectively, are not statistically independent when estimated from the same data.

SUMMARY OF DISTRIBUTION CONCEPTS FROM CHAPTER 2

The life-table estimator smooths by computing the hazard rate for a time interval rather than at a time point. Alternatively, smoothing algorithms may be applied to estimates of the hazard rate at a set of adjacent time points in a manner analogous to computing a running mean. The normal distribution is very commonly encountered in science. It is obvious that if a random variable X is normally distributed, functions of that random variable (such as X2,X3,logX, and 1-X) will usually not be normally distributed.

Therefore, the class width is indeed the same as the class size. The probability of y successes, then, is obtained by repeated application of the addition rule. That is, the probability of y successes is obtained by multiplying the probability of a sequence by the number of possible sequences, resulting in the above formula.

4 Distributions

A frequency distribution of the variable price is shown in Table 1.6. Clearly the preponderance of homes is in the 50- to 150-thousand-dollar range. To provide more information, we will construct frequency distributions by grouping the data into categories and counting the number of observations that fall into each one. Because we want to count each house only once, these categories (called classes) are constructed so they don’t overlap. Because we count each observation only once, if we add up the number (called the frequency) of houses in all the classes, we get the total number of houses in the data set. Nominally scaled variables naturally have these classes or categories.

In the case of radioactive decay, the memoryless property corresponds to the fact that the decay rate (the probability of decay per unit time interval) is independent of the age of the nuclei. If the failure rate of a component is modeled by an exponential distribution, the memoryless property corresponds to the failure being independent of age, i.e., the component shows no wear and tear due to its age. Failure rates that do not have the memoryless property are discussed in the section on the Weibull distribution. Table II shows some important special cases of this cumulative distribution function. Approximately 68% of the values in any normal population lie within one standard deviation (σ) of the mean µ, approximately 95% lie with two standard deviations of µ, and approximately 99.7% lie within three standard deviations of µ. Another presentation of a distribution is provided by a pie chart, which is simply a circle (pie) divided into a number of slices whose sizes correspond to the frequency or relative frequency of each class.

The total number of data items with a value less than or equal to the upper limit for the class i

The parameter β is a shape parameter affecting the shape of the distribution, while α is a scaling parameter affecting the scale. It is also possible to change the location of the Weibull distribution by replacing the variable x by x-x0. Repeated physical measurements, such as measurements of the length of an object, are usually normally distributed. Data from any distribution (whatever its form) have a mean whose distribution is approximately normal.

The mean of a probability distribution is often called the expected value of the random variable. For example, the expected number of individuals in a couple who have had measles is 0.4. This is a “long-range expectation” in the sense that if we sampled a large number of couples, the expected (average) number of individuals who have had measles would be 0.4. Note that the expected value can be (and often is) a value that the random variable may never attain.

constructing frequency distribution;, a5 the number of classes is decreased, the class width

An integer related to sample size that determines which member of a family of probability distributions is appropriate in a particular case. The standard normal has only one member and does not use df, but the t has many members and does use df. A relative frequency distribution consists of the relative frequencies, or proportions (percentages), of observations belonging to each category.