-
bimodal distribution
If data have exactly two modes
-
multimodal distribution
If data have more than two modes
-
arithmetic mean
- An important property of the arithmetic mean is that the sum of the deviations from the
- mean will always equal 0.
- arithmetic mean is sensitive the to extreme values. We often refer to these extreme values
- (whether small or large) as outliers.
-
Organization of Frequency and Relative Frequency
Distributions
- Step 1: Determine the number of class intervals (also referred
- to as classes or categories, and meaning ranges of data values)
- of interest. Between 5 and 20 class intervals are generally recommended. Class intervals
- should be selected so that each data point (or individual value of raw data) can fall into
- only one category.
- Step 2: When it is desired that all class intervals be of equal
- width, you determine the width of the class by subtracting the smallest data value from
- the largest and then dividing by the number of class intervals desired.
-
Weighted Mean
If you don't want to weigh the data points equally, but by their relative importance.
- (95 0.10) +(70 0.10)+ (60 0.10)+ (85 0.30)+ (90 0.40)= 84 (weighted mean)
-
dispersion or variability
-
range
... is one measure of dispersion of a data set.
range = largest data value – smallest data value
- The problem with the range is that it includes only two numbers of the data set and
- ignores the rest of the values.
-
Percentiles
- Percentiles of a ranked data set divide it into hundredths, or 100 equal parts
- of the data values. The median is the 50th percentile. Fifty percent of the data falls
- below the median and 50 percent falls above it.
- Quartiles are the percentiles that divide the data into quarters (or fourths).
- There are three quartiles, then, at the 25th, 50th, and 75th percentiles. We often refer
- to these as Q1, Q2, and Q3, respectively.
-
How we calculate percentiles:
- 1. Arrange the data in ascending order from the smallest value to the largest.
- 2. Compute the index i:
- where
- i is the position number of the percentile you're interested in
- p is the percentile you're interested in knowing
- n is the number of items in the data set
- 3. If i is not an integer, round up to the nearest integer. The next integer value
- greater than i
- denotes the position of the pth percentile.
- If i is an integer, the pth percentile is the average of the data values
- in positions i and i + 1.
-
Inter-quartile Range
- The inter-quartile range is the 75th percentile minus the 25th percentile, or Q3 –
- Q1. This range has less dependency on outliers than does the range previously discussed.
-
variance of a data set
- is an important measure of dispersion
- within a data set because it takes into account all the data values
-
The variance of the population
- is the average of the squared deviations from the
- arithmetic mean. When you take the variance of a sample, you divide the squared deviations
- from the sample mean by the sample size minus 1. Doing this generally gives a better
- estimate of the population variance from which the sample comes.
-
denote the variance of a population with...
"little sigma squared."
-
We denote the sample variance
s 2 (squared)
-
The population mean and the population variance are called
- parameters of a
- population because they are quantities that are fixed for any given population.
-
We
call the sample mean and the sample variance
- sample statistics (or random
- variables) because they vary from one sample to another, inasmuch as their values depend
- on which sample is selected.
-
Use the following steps to calculate the sample variance:
- 1. Calculate the sample mean.
- 2. Calculate the difference between each observation and the sample mean.
- 3. Square each difference found in step 2.
- 4. Sum the squared differences found in step 3.
5. Divide the sum of the squared differences by the sample size minus one, n – 1.
-
Frequently, we use the ________ _________ instead of the variance to
describe dispersion.
- the standard deviation. You get the standard deviation by taking the square root of the
- variance. ( sample variance = s squared/population variance = "little sigma squared")
- The advantage of using the standard deviation is that it has the same units of
- measurement as the data values.
-
We
represent this statistic with s, meaning "the square root of the
variance, s2 (squared)."
This is the representation for standard deviation.
-
What does the standard deviation actually mean?
- The standard deviation shows how the
- data points are distributed or dispersed about the sample mean. When the things you are
- measuring are alike, such as test scores from the same class, the bigger the standard
- deviation, the more dispersion you have about the mean.
-
Coefficient of Variation
- When two (or more) distributions have the same mean, the one with the largest standard
- deviation has the most variation. But what about when distributions have different means?
- In that case, you can't compare just the standard deviations. Instead, you have to compare
- the coefficient of variation (CV) for each distribution as well. The distribution with the
- highest CV has the most dispersion.
-
Empirical Rule
- This rule applies to data that are approximately normally distributed, that is, a
- bell-shaped symmetrical distribution. About 68 percent of the data points will fall within
- one standard deviation of the mean, and about 95 percent of the data points will be within
- two standard deviations of the mean.
- For example, let's continue with our inquiry into salaries but with a different
- profession. Let's take a sample of the salaries of 150 production workers. Here's a
- distribution of salaries we might find.
-
Chebyshev's Theorem
- tells us
- the minimum proportion of data points that lie within any number of standard deviations
- from the mean, regardless of the shape of the distribution. Chebyshev's theorem states:
- At least of
- the measurements fall within k standard deviations from the mean.
- Note: k must be greater than 1.
- For example, if you want to find out the minimum percentage of the data values that are
- within 2 standard deviations from the mean, you'd calculate:
- That is, for any data set, at least 75 percent of the data values are within two
- standard deviations from the mean.
- If you calculate the minimum percentage of values are between the mean and three
- standard deviations from the mean, you'll get an answer of "at least 89
- percent."
- Although Chebyshev's theorem provides us only with lower bounds for the percentage of
- data values that lie within k (where k >1) standard deviations from the
- mean, it doesn't provide us with exact percentages. The power of Chebyshev's theorem lies
- in the fact that it is true for any distribution, regardless of its shape.
-
The ____ _ ______ compares the standard deviation relative
to the mean of the distribution. For this reason, the CV is also known as the ______ ______ _____ (RSE).
- coefficient of variation;
- relative standard error
-
Here's how we calculate the CV...
- Think of the CV for any variable as the precision of the mean for that variable. Many
- federal agencies, such as the National Center for Health Statistics (NCHS), use the CV as
- a measure of the precision or reliability of estimates of health characteristics. The
- smaller the CV, the more reliable (precise) the estimate is. The larger the CV, the more
- unreliable it is.
-
Shapes of distributions
- 1. Symmetrical distributions- Has the same center value for the mean, median, and mode.
- (mirrored appearance)
2. Uniform of Rectangular Distribution- Every class has the same frequency.
3. Skewed Distribution- One "tail" is longer than the other.
- If the longer tail is on the left, we say that the distribution is skewed to the
- left, or negatively skewed. If the longer tail is to the right, we say the
- distribution is skewed to the right, or positively skewed.
- 4. Bimodal Distribution- A bimodal distribution refers to a histogram in which two classes with largest
- frequencies are separated by at least one class, and the top two frequencies of these
- classes may have different values.
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