AP STATS- Chapter 2
Standard Normal Distribution Equation
*if a variable x has any normal distribution N(μ, σ) with mean μ and standard deviation σ, then the standardized variable has the standard normal distribution N (0,1).
Effect of Adding (or Subtracting) a Constant
- changes measures of center and location (mean, median, quartiles, percentiles) - doesn't change the shape of the distribution or measures of spread (IQR, range, standard deviation)
Aspects of a Density Curve
- describes the overall pattern of a distribution. the area under the curve and above any interval of values on the horizontal axis is the proportion of all observations that fall in that interval. -come in many shapes. -often a good description of a distribution. -outliers are not described by the curve.
Rules to Making a Cumulative Relative Frequency Graph
- it is customary to start a cumulative relative frequency graph with a point at a height of 0% @ the smallest value of the 1st class. the last point we should plot should be at a height of 100%. we connect consecutive points with a line segment to form the graph. equally spaced intervals!
Effect of Multiplying (or Dividing) By a Constant
-changes center, location (mean, median, quartiles, percentiles), measures of spread (range, IQR, standard deviation). - doesn't change shape of the distribution.
Exploring Quantitative Data
1) always plot your data. graphs. 2) look for overall pattern (shape, center, spread) and for striking departures (outliers) 3) calculate numerical summaries to describe center and spread. 4) sometimes the overall pattern of a large # of observations is so regular that we can describe it by a smooth curve. (adjust the scale of the graph so that the total area under the curve is exactly 1)
Why are Normal Curves Important?
1) good descriptions for distributions of real data. 2) good approximations to the results of many kinds of chance outcomes. 3) many statistical procedures (inference) are based on normal distributions. (even though many sets of data follow a normal distribution, many do not)
How to Find Areas in Any Normal Distribution
1) state the distribution and the values of interest. draw a normal curve with the area of interest shaded and the mean, standard deviation, and boundary value(s) clearly identified. 2) perform calculations- show your work! do one of the following: (i) compute a z-score for each boundary value and use table A or technology to find the desired area under the standard normal curve (ii) use the normalcdf command and label each of the inputs. 3) answer the question!
How to Find Values From Areas in Any Normal Distribution
1) state the distribution and the values of interest. draw a normal curve with the area of interest shaded and the mean, standard deviation, and unknown boundary value clearly identified. 2) perform calculations- show your work! do one of the following: (i) use table A or technology to find the value of z with the indicated area under the standard normal curve, then "unstandardized" to transform back ot the original distribution, (ii) use the invNorm command and label each of the inputs. 3) answer the question!
Formula for Celsius and Fahrenheit
C=5/9(F-32)
The 68-95-99.7 Rule (Empirical Rule)
In the Normal distribution with mean μ and standard deviation σ, approximately 68% of the observations fall within σ of the mean μ, approximately 95% of the observations fall within 2σ of μ, and approximately 99.7% of the observations fall within 3σ of μ.
How Do We Abbreviate Normal Distribution?
N(mean, standard deviation)
How Do We Standardize a Value?
Subtract the mean of the distribution and then divide the difference by the standard deviation.
Density Curve
a curve that is always on or above the horizontal axis and has area of exactly 1 underneath it.
Standardizing
converting observations from original values to standard deviation units.
Normal Distribution
distribution described by a normal density curve. any particular normal distribution is completely specified by two numbers, its mean and standard deviation. symmetric= mean same as median.
Cumulative Relative Frequency Graph
graph used to examine location within a distribution.
Notation for Mean of Density Curve
greek letter "mu" (looks like a u but with a long tail at the front)
Notation for Standard Deviation of Density Curve
greek letter "sigma" (looks like o but with a long tail at the end)
What Does the Z-Score Tell Us?
how many standard deviations from the mean an observation falls, and in what direction. (greater= positive, less than= negative)
Normal Probability Plot
if the points on this plot lie close to a straight line, the data are approximately normally distributed. systematic deviations from a straight-line indicate a non-normal distribution. outliers appear as points that are far away from the overall pattern of the plot. *when you examine a normal probability plot, look for shapes that show clear departures from normality. don't overreact to minor wiggles in the plot.
Standardized Score (z-score)
if x is an observation from a distribution that has a known mean and standard deviation, the standardized score is (x-mean)/standard deviation.
Rule about Quantitative Data
if you start with ANY set of quantitative data and convert these values to standardized scores (z-scores), the transformed data set will have a mean of 0 and a standard deviation of 1. (shape= same!)
Normal Curves
important class of density curves that are symmetric, single-peaked, and bell-shaped.
Chebyshev's Inequality
in any distribution, the proportion of observations falling within k standard deviations of the mean is at least 1-(1/k^2).
What is Special About Density Curves?
no set of real data is exactly described by this curve. it is an approximation that is easy to use and accurate enough for practical use.
______ and ______ can compare the relative location of individuals in different distributions.
percentiles, z-scores
The ____ and ____ Alone DO NOT Specify the Shape of Most Distributions.
standard deviation and mean.
The Standard Normal Table (A)
table A 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.
Median of a Density Curve
the "equal areas point", the point with half the area under the curve to its left and the remaining half of the area to its right. because density curves are idealized patterns, a symmetric density curve is exactly symmetric (mean and median are equal) median= center!
Mean of a Density Curve
the balance point at which the curve would balance if made of solid material.
What is the Mean of a Normal Distribution?
the center of the symmetric normal curve.
What is the standard deviation of a normal distribution?
the distance from the center to the change of curvature points on either side.
What Pulls the Mean?
the long tail.
Standard Normal Distribution
the normal distribution with a mean of 0 and a standard deviation of 1.
What are Located at a Distance of Standard Deviation on Either Side of the Mean?
the points at which the change of curvature takes place.
Percentile
the pth percentile of a distribution is the value with p percent of the observations less than it or equal to it.
What is the Natural Measure of Spread for Normal Distributions?
the standard deviation.
What Controls the Spread of the Normal Curve?
the standard deviation. curves with larger SD are more spread out.
How does a Cumulative Relative Frequency Graph work?
they begin by grouping the observations into equal-width classes. the completed graph shows the accumulating percent of observations as you move through the classes in increasing order. (specific percentiles)
How Does Changing the Mean Without Changing the Standard Deviation Affect the Graph?
this moves the normal curve along the horizontal axis without changing its spread.
What is a Common Mistake When Using Table A?
to look up a z-value in table A and report the entry corresponding to that z-value, regardless of whether the problem asks for the area to the left or to the right of that z-value. to prevent this, always sketch the standard normal curve, mark the z-value and shade the area of interest.