Chapter 2 The Normal Distributions “Statistically thinking will one day be as necessary for efficient citizenship as the ability to read and write.” H. G. Wells

2.1 Density Curves and the Normal Distributions We can sometimes use a mathematical model as an approximation to the overall pattern of data. Examples: Normal distributions (Ch. 2) Linear regression line (Ch. 3)

Density Curve Displays the overall pattern or shape of a distribution. Has an area of exactly 1 square unit underneath of it. Is on or above the horizontal axis. The median is the point that divides the area under the curve into halves. The mean is the “balance point” of the curve. (think: teeter totter)

Normal Distributions (cont’d) Data that display seen in Figure w.w occur very frequently. Theoretical mathematical models used to approximate such distributions are called normal distributions. Normal distributions are formed by a special type of density curves. NB. Use of plural  there is NOT just one normal curve! There is a WHOLE FAMILY of them!!!

Normal Distributions (cont’d) Each member of the family of normal distributions share THREE distinguishing characteristics:  Every normal distribution is symmetric.  Every normal distribution has a single peak at its center.  Every normal distribution follows a bell- shaped curve.

Normal Distributions (cont’d) Each member of the family of normal distributions is identified two things: Mean Symbol: Determines where the center is Peak of a normal curve Point of symmetry Standard Deviation Symbol: Indicates how spread out the distribution is Equals the distance between mean and the points where the curvature changes

Normal Distributions (cont’d) Can use normal curves to calculate (approximately) the proportion of a population’s observations that fall in a given interval of values. This proportion is also known as the probability that the value of a particular member of the population will fall in the given interval. Geometrically, this proportion (or probability) is equivalent to finding the area under the curve over the given interval. The TOTAL AREA UNDER THE CURVE is ALWAYS equal to 1 (or 100%)

Normal Distributions (cont’d) In the normal distribution: 68% of the observations fall within 1 standard deviation of the mean. 95% of the observations fall within 2 standard deviations of the mean. 99.7% of the observations fall within 3 standard deviations of the mean.

Graphically (TI – 83) normalcdf(lowerbound, upperbound, mean, standard deviation) can be very useful in statistical analysis. If normal distribution has mean = 0 and st. dev. = 1, then Normalcdf(-1,1,0,1)= Normalcdf(-2,2,0,1)= Normalcdf(-3,3,0,1)=

Graphically (TI – 83) Window: xmin = -4, xmax = 4 ZoomFit Gives you a normal distribution curve with mean = 0 and st.dev. = 1.

Percentile Percent of distribution that is at or the the left of the observation. Example: a test score representing the 80 th percentile means that 20% of the test takers scored higher!

2.2 Standard Normal Calculations “He uses statistics like a drunken man uses a lamppost…for support rather than illumination.” Andrew Lang

Standardized Value Sometimes called a z-score. Formula to find the z-score of an observation x : A z-transformation changes a normal random variable with mean and standard deviation into a standard normal random variable with mean = 0 and standard deviation = 1.

Try It!! Complete the table: x z-score

Solution Complete the table: x z-score 5 z = z = z = z = z = 1.44

Standard Normal Distribution Standardizing a variable that has any normal distribution produces a new variable that has the standard normal distribution. Symbolism Variable x has a normal distribution with mean and standard deviation N(0,1) means standardized distribution is normal and has a mean = 0 and st. dev. = 1. Standard normal table is an important table to be able to use!

Calculations with Normal Distributions Any question about what proportion of observations lie in some range of values can be answered by finding an area under the curve. We use Table A in the back of the book. It is a table of areas under the standard normal curve. The table entry for each value z is the area under the curve to the left of z.

Finding Normal Proportions State the problem in terms of the observed variable x. Standardize x to restate the problem in terms of a standard normal variable z. Draw a picture to show the area under the curve. Find the required area under the standard normal curve (use Table A and the fact that the total area under the curve is 1.)