Sasha Thackaberry's List: Statistics

• Variables
• Variables are properties or characteristics of  some event, object, or person that can take on different values  or amounts (as opposed to constants such as π   that do not vary).
• When  a variable is manipulated by an experimenter, it is called an independent  variable
• In general the independent variable is manipulated  by the experimenter and its effects on the dependent variable  are measured.
• If an experiment compares an experimental treatment   with a control treatment, then the independent variable (type   of treatment) has two levels: experimental and control
• In general, the number   of levels of an independent variable is the number of experimental   conditions.
• Qualitative variable are those that express a  qualitative attribute such as hair color, eye color, religion,  favorite movie, gender, and so on
• Quantitative variables  are those variables that are measured in terms of numbers
• Discrete and Continuous   Variables
• discrete   variables since the possible scores are discrete points on   the scale
• are continuous   variables since the scale is continuous and not made up of   discrete steps
• • continous means also its unbounded by a definite scale with a end point.

  • continuous variables could be any number (with decimals, etc., this sets it apart from "discrete variables"

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• Variables that can only take on a finite number of values are   called "discrete variables." All qualitative   variables are discrete. Some quantitative   variables are discrete, such as performance rated as 1,2,3,4, or 5, or temperature   rounded to the nearest degree. Sometimes, a variable that takes on enough discrete   values can be considered to be continuous for practical purposes. One example   is time to the nearest millisecond.
   
    Variables that can take on an infinite number of possible values are called   "continuous variables."

• Distributions

• This table is called a frequency   table and it describes the distribution of M&M color frequencies.   Not surprisingly, this kind of table is called a frequency   distribution. Often a frequency distribution is shown graphically   as in Figure 1.

   
• Continuous Variables

• The variable "color of M&M" used in   this example is a discrete   variable, and its distributions is also called discrete.   Let us now extend the concept of a distribution to continuous   variables.

   
• The solution to this problem is to create a grouped  frequency distribution. In a grouped frequency  distribution, scores falling within various ranges are tabulated.  Table 3 shows a grouped frequency distribution for these  20 times.

• Grouped frequency distributions can be portrayed   graphically. Figure 3 shows a graphical representation of the   frequency distribution in Table 3. This kind of graph is called   a histogram.   A later chapter contains an entire section devoted to histograms.

     
• Probability Densities
• To represent the probability  associated with an arbitrary movement (which can take any positive  amount of time), we must represent all these potential times  at once. For this purpose, we plot the distribution for the  continuous variable of time. Distributions for continuous variables  are called   continuous distributions. They also  carry the fancier name probability  density. Some probability densities have particular  importance in statistics. A very important one is shaped like  a bell, and called the normal  distribution
• The normal distribution  doesn't represent a real bell, however, since the left and  right tips extend indefinitely
• Shapes of Distributions
• A distribution with the longer tail extending in the positive  direction is said to have a positive  skew. It is also described as "skewed to the  right."
• Although less common, some distributions have negative  skew. Figure 8 shows the scores on a 20-point problem  on a statistics exam. Since the tail of the distribution  extends to the left, this distribution is skewed  to the left.
• A distribution with two peaks is called a bimodal  distribution.
• The  upper distribution has relatively more scores in its tails; its  shape is called leptokurtic.  The lower distribution has relatively fewer scores in its tails;  its shape is called platykurtic
• The distribution of empirical data is called a frequency distribution and consists of a count of the number of occurences of each value.
• In a symmetric distribution, the tails are equally long. However, the tails themselves are not skew. Skew is defined as the relative lengths of the tails. A distribution cannot have both negative and positive skew.

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• Summation Notation
• Many statistical formulas involve summing numbers.   Fortunately there is a convenient notation for expressing summation.   This section covers the basics of this summation   notation.

• Linear Transformations
• Notice that the points form a   straight line. This will always be the case if the transformation   from one scale to another consists of multiplying by one constant   and then adding a second constant. Such transformations are therefore   called linear   transformations.
• Many transformations are not linear. With nonlinear transformations, the points in a plot of the transformed variable against the original variable would not fall on a straight line. Examples of nonlinear transformations are: square root, raising to a power, logarithm, and any of the trigonometric functions.

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• Graphing Qualitative Variables
• The key point about the qualitative   data that occupy us in the present section is that they do not   come with a pre-established ordering (the way numbers are ordered
• Frequency Tables
• • the relative frequency always adds up to "1" in a frequency table.
• the relative frequencies,   which are the proportion of responses in each category
• Pie Charts
• Pie charts are effective for displaying the relative   frequencies of a small number of categories. They are not recommended,   however, when you have a large number of categories.
• Here is another important point about pie charts.   If they are based on a small number of observations, it can be   misleading to label the pie slices with percentages
• Bar charts
• Bar charts can also be used to represent frequencies   of different categories
• Comparing Distributions
• The bars in Figure 3 are oriented horizontally   rather than vertically. The horizontal format is useful when you   have many categories because there is more room for the category   labels
• Some graphical mistakes to avoid
• Edward   Tufte coined the term "lie factor" to refer to the   ratio of the size of the effect shown in a graph to the size of   the effect shown in the data. He suggests that lie factors greater   than 1.05 or less than 0.95 produce unacceptable distortion.
• Another distortion in bar charts results from   setting the baseline to a value other than zero
• that it is a serious mistake   to use a line graph when the X-axis contains merely qualitative   variables. A line graph is essentially a bar graph with the tops   of the bars represented by points joined by lines (the rest of   the bar is suppressed
• Pie charts and bar charts can both be effective   methods of portraying qualitative data. Bar charts are better   when there are more than just a few categories and for comparing   two or more distributions. Be careful to avoid creating misleading   graphs

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• Quantitative
• Variables
• • this is an ezxMPLE sticky
• quantitative variables   are variables measured on a numeric scale. Height, weight, response   time, subjective rating of pain, temperature, and score on an exam   are all examples of quantitative variables
• The upcoming  sections cover the following types of graphs: (1) stem and leaf  displays, (2) histograms, (3) frequency polygons, (4) dot plots,  (5) box plots, and (6) scatterplots
• Some graph types such as  stem and leaf displays are best-suited for small to moderate  amounts of data whereas others such as histograms are best-suited  for large amounts of data. Graph types such as box plots are  good at depicting differences between distributions. Scatterplots  are used to show the relationship between two variables

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• Histograms
• A histogram is a graphical method for displaying   the shape of a distribution. It is particularly useful when there   are a large number of observations.
• Grouped Frequency Distribution
• called     class   intervals
• the   class   frequencies
• Class intervals of width 10 provide enough detail   about the distribution to be revealing without making the graph   too "choppy
• Placing   the limits of the class intervals midway between two numbers (e.g.,   49.5) ensures that every score will fall in an interval rather   than on the boundary between intervals.
• The histogram makes it plain that most of the scores   are in the middle of the distribution, with fewer scores in the   extremes. You can also see that the distribution is not symmetric:   the scores extend to the right farther than they do on the left.   The distribution is therefore said to be   skewed
• Histograms can also be used when the scores are measured on a   more continuous scale such as the length of time (in milliseconds)   required to perform a task. In this case, there is no need to   worry about fence sitters since they are improbable. (It would   be quite a coincidence for a task to require exactly 7 seconds,   measured to the nearest thousandth of a second
• Using   whole numbers as boundaries avoids a cluttered appearance, and   is the practice of many computer programs that create histograms.   Note also that some computer programs label the middle of each   interval rather than the end points
• Histograms can be based on relative   frequencies instead of actual frequencies. Histograms based   on relative frequencies show the proportion of scores in each   interval rather than the number of scores
• You can change a histogram based on frequencies   to one based on relative frequencies by (a) dividing each class   frequency by the total number of observations, and then (b) plotting   the quotients on the Y axis (labeled as proportion).
• The best advice is to experiment with different choices of width,   and to choose a histogram according to how well it communicates   the shape of the distribution
• Bin width is another name for the width of each class interval

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• Box Plots
• Box plots are useful for identifying   outliers and for comparing distributions
• outside values are indicated by small   "o's, and far out values are indicated by asterisks.
• Box plots provide basic information about a distribution. For example, a distribution with a positive skew would have a longer whisker in the positive direction than in the negative direction. A larger mean than median would also indicate a positive skew
• Box plots are good at portraying extreme values and are especially good at showing differences between distributions
• The H-spread is the difference between the upper hinge (D) and the lower hinge (G).
• The median is represented by a line through the middle of the box.

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• Line Graphs
• A line graph is a bar graph with the tops of the   bars represented by points joined by lines (the rest of the bar   is suppressed
• Line graphs are appropriate only when both the X- and Y-axes display ordered (rather than qualitative) variables
• line graphs are generally better at comparing changes over time
• Let us stress that it is misleading to use a line graph when the X-axis contains merely qualitative variables

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• Summarizing Distributions

• Central Tendency
• Central tendency is a loosely defined concept that has to do  with the location of the center of a distribution

• Measures of Central Tendency
• In the previous section we saw that there are  several ways to define central tendency. This section defines  the three most common measures of central tendency: the mean,  the median, and the mode
• Arithmetic Mean
• The arithmetic mean is the most common measure  of central tendency. It is simply the sum of the numbers divided  by the number of numbers
• The symbol "μ" is used for the  mean of a population. The symbol "M" is used for the mean of  a sample. The formula for μ is shown below
• μ = ΣX/N
    where ΣX is the sum of all the numbers in the population   and
    N is the number of numbers in the population
• The formula for M is essentially identical
• M = ΣX/N
    where ΣX is the sum of all the numbers in the sample and
    N is the number of numbers in the sample.
• Median
• The median is the   midpoint of a distribution: the same number of scores are above   the median as below it
• The median can also be thought of as the   50th percentile.
• When there is an odd number of numbers, the median   is simply the middle number
• When there is an even number of numbers, the median   is the mean of the two middle numbers
• The mode is the most frequently occurring value.
• With continuous, data such as response time measured to many decimals,   the frequency of each value is one since no two scores will be   exactly the same (see discussion of continuous   variables). Therefore the mode of continuous data is normally   computed from a grouped   frequency distribution.

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• Measures of Variability
• Variability refers to how "spread out"   a group of scores is.
• Just  as in the section on central tendency we discussed measures  of the center of a distribution of scores, in this chapter we  will discuss measures of the variability of a distribution
• The terms variability, spread, and dispersion are  synonyms, and refer to how spread out a distribution is
• There are four frequently used measures of variability, the  range: interquartile range, variance, and standard deviation
• • re:  range, the other book says it is the difference plus 1 (so the first example would be 10 - 8 + 1 = 3
• Range
• The range is the simplest measure of variability  to calculate, and one you have probably encountered many times  in your life. The range is simply the highest score minus the  lowest score.
• Interquartile Range
• IQR = 75th percentile - 25th percentile
• Recall that in the discussion of box  plots, the 75th percentile  was called the upper hinge and the 25th percentile was called  the lower hinge. Using this terminology, the interquartile range  is referred to as the H-spread
• Variability can also be defined in terms of how   close the scores in the distribution are to the middle of the   distribution. Using the mean as the measure of the middle of the   distribution, the variance is defined as the average squared difference   of the scores from the mean
• Standard Deviation
• The standard deviation is an especially  useful measure of variability when the distribution is normal  or approximately normal (see  Chapter 6) because the proportion of the distribution  within a given number of standard deviations from the mean  can be calculated

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• Shapes of Distributions
• Distributions with positive skew normally have  larger means than medians

• Additional Measures of Central Tendency
• Although the mean, median, and mode are by far   the most commonly used measures of central tendency, they are   by no means the only measures. This section defines three additional   measures of central tendency: the trimean, the geometric mean,   and the trimmed mean
• Trimean
• The trimean is a weighted   average of the 25th percentile, the 50 percentile, and the 75th   percentile
• Trimean = (P25 + 2P50 +   P75)/4
• the median is   weighted twice as much as the 25th and 75th percentiles
• Geometric Mean
• The geometric mean is computed by multiplying   all the numbers together and then taking the nth root of the product
• the equation says to multiply all the values of X and then raise   the result to the 1/Nth power
• Raising a value to the 1/Nth power   is, of course, the same as taking the Nth root of the value
• The question is how to compute annual rate of  return? The answer is to compute the geometric mean of the returns.
• Trimmed Mean
• To compute a trimmed mean,   you remove some of the higher and lower scores and compute the   mean of the remaining scores
• A mean trimmed 10% is a mean computed   with 10% of the scores trimmed off; 5% from the bottom and 5%   from the top

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• Comparing Measures of Central Tendency
• How do the various measures of central tendency   compare with each other? For symmetric   distributions, the mean, median, trimean, and trimmed mean   are equal, as is the mode except in bimodal   distributions. Differences among the measures occur with skewed   distributions
• When  distributions have a positive skew,  the mean is typically higher than the median, although it may  not be in bimodal distributions
• Fortunately,  there is no need to summarize a distribution with a single number.  When the various measures differ, our opinion is that you should  report the mean, median, and either the trimean or a the mean  trimmed 50%. Sometimes it is worth reporting the mode as well.

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• Areas Under Normal Distributions
• Figure 2 shows a normal distribution with a mean   of 100 and a standard deviation of 20. As in Figure 1, 68% of   the distribution is within one standard deviation of the mean.
• The normal distributions shown in Figures 1 and   2 are specific examples of the general rule that 68% of the area   of any normal distribution is within one standard deviation of   the mean.
• For all normal   distributions, 95% of the area is within 1.96 standard deviations   of the mean. For quick approximations, it is sometimes useful   to round off and use 2 rather than 1.96 as the number of standard   deviations you need to extend from the mean so as to include 95%   of the area.

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• History of Normal Distribution
• Abraham de Moivre, an 18th century statistician   and consultant to gamblers was often called upon to make these   lengthy computations. de Moivre noted that when the number of   events (coin flips) increased, the shape of the binomial distribution   approached a very smooth curve
• The importance of the normal curve stems primarily   from the fact that the distribution of many natural phenomena   are at least approximately normally distributed
• This same distribution had been discovered by Laplace   in 1778 when he derived the extremely important central   limit theorem,
• Laplace showed that even if a distribution  is not normally distributed, the means of repeated samples from  the distribution would be very nearly normal, and that the larger  the sample size, the closer the distribution would be to a normal  distribution
• Inferential statistics can only be computed with a normal distribution
• Laplace derived the central limit theorem. Adrian and Gauss developed the formula for the normal distribution.

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