Biostatistics in Dentistry

Transcription

1 Biostatistics in Dentistry Continuous probability distributions Continuous probability distributions Continuous data are data that can take on an infinite number of values between any two points. Examples of types of data usually considered continuous: weight, length, time 1

2 Probability density function We can describe continuous data using a probability density function The area under the curve between two points on the horizontal axis signifies the probability of an observation falling between those two points. Probability density function Probabilities are associated with intervalsrather than single points. Probability of a single point is zero P(X = b) = 0, thus P(a < X < b) = P(a < X <b) 2

3 Probability, mean, and variance formulas For a probability density function f(x), = P ( a < X < b) f ( x) dx, E( X ) = µ = Var( X ) = b a xf ( x) dx, ( x µ ) 2 f ( x) dx, Example:Probability density function for mean attachment level mean attachment level (mm) Estimated from NHANES III data Attachment level was assessed by measuring attachment levels at the mesial and buccalaspects of all teeth in randomly assigned half-mouths, and averaging these values. 3

4 Example:Probability density function for mean attachment level mean attachment level (mm) Estimated from NHANES III data This graph shows that most of the mass (most of the people in the population) lies between 0 mm and 2mm, with less and less likelihood of higher and higher values. The histogram of a sample is an estimate of the population probability density function. sample of 100 patients mean attachment level (mm) A histogram describes the sample of data points the same way that a probability density function describes the entire population 4

5 The Normal distribution Many forms of data follow a Normal distributionor can be transformed to be approximately Normal. Sometimes referred to as a Gaussian distribution. For almost any type of data, sums and averages of repeated observationswill be well-described by Normal distributions Normal probability distribution The Normal density function is characterized y a bell shaped curve Symmetric about the mean, µ Has 68% its area within 1 standard deviation of μ. Has 95% its area within 2 standard deviations of μ. Normal distribution µ - 3σ µ - 2σ µ - σ µ µ + σ µ + 2σ µ + 3σ 5

6 Normal distribution notation N(μ, σ 2 ) is used to denote a Normal distribution with mean μ and variance σ 2 (standard deviation σ). A Standard Normal distribution is a Normal distribution with µ= 0 and σ 2 = 1. Also denoted N(0,1), and also often by Z. Normal distribution µ - 3σ µ - 2σ µ - σ µ µ + σ µ + 2σ µ + 3σ Standard Normal distribution Computing standard Normal probabilities Table 3 in the coursepack(pp177-8) can be used to compute standard Normal probabilities. The Excel function NORM.S.DIST can also be used to compute standard Normal probabilities. 6

7 Example: Using Table 3 to compute a probability P(Z < 0.54) =? P(Z > 0.54) = = Computing standard Normal probabilities Compute: P(1.00 < Z <1.42) From Table 3 (or Excel): P(Z < 1.42) = P(Z < 1.00) = P(1.00 < Z <1.42) = P(Z <1.42) P(Z <1.00) = =

8 General Normal distribution probabilities Useful property of Normal distributions If X ~ N(µ, σ 2 ), then (X -µ)/σ~ N(0, 1) So, if X has a N(µ, σ 2 ) distribution, then we can convert the general Normal probability into a standard Normal probability P P And now we can use the standard Normal tables in order to find the probability Example: converting a general Normal probability to a standard Normal probability Suppose X is Normally distributed with mean 2.3 and standard deviation 0.6. Find P(X > 2). P 2 P P 0.5 P Note that P(Z> -0.5) = P(Z< 0.5) by the symmetry of the Normal distribution 8

9 Finding percentiles of Normal distributions Example: What is the 90 th percentile of a N(0,1) distribution? Need to find asuch that P(Z< a) = Can use Table 3 in coursepack. Work backward by finding the values closest to 0.90 in the column labeled P(Z < z). Finding percentiles of Normal distributions The row with 0.90 in the P(Z < z) column has 1.28 in the z column. Thus P(Z < 1.28) = The 90 th percentile of a standard Normal distribution is Z.90 = 1.28 The box at bottom right gives selected percentiles, for an easier way to look up commonly-used percentiles 9