![]() On a smaller note, in a pinch (stuck with only a calculator and a few remembered values from the normal table) I have quite successfully applied Simpson's rule (and related rules for numerical integration) to get a good approximation at other values it's not all that tedious to produce an abbreviated table* to a few figures of accuracy. So rather than thousands of Riemann sums it was hundreds of Taylor expansions. ![]() $$\begin$, so that its omission is justified.ĭavid indicates that the tables were widely used. The two graphs below show the area under the Z distribution represented by P(Z < z 1).Laplace was the first to recognize the need for tabulation, coming up with the approximation: CumulativeĪ cumulative Z table provides the probability that a statistic is less than Z, or P(Z < z 1). As such, it is important to be aware of the type of Z table being used. As can be seen, there are a number of ways to determine various probabilities using Z tables. Alternatively, this probability can be directly read off a complementary cumulative Z table (described below). For example, to determine P(Z > z 1) subtract P(0 z 1). Therefore, P(Z 1.5), and can be found by subtracting P(Z z 1), by adding or subtracting the appropriate areas under the curve. Since 50% of values lie above and below the mean, the probability that a score will be below an 87 is represented by the following distribution: However, this probability only represents the probability from the mean to the Z-score, as shown in the figure below: Referencing the above Z table, a Z-score of 1.5 corresponds to a probability of 0.43319, or around 43%. Use a cumulative from mean Z table to find the probability of a score being above or below an 87. The average score on a math exam for a class of 150 students was a 78/100 with a standard deviation of 6. The x-value (and all values in the normal distribution) is standardized as follows: The figure below shows a normal distribution with μ = 5, σ = 4, and x = 11, as well as its corresponding Z distribution.
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