Sampling in which the data is collected using chance methods or random numbers. Systematic Sampling Sampling in which data is obtained by selecting every k th object. Convenience Sampling Sampling in which data is which is readily available is used. Stratified Sampling Sampling in which the population is divided into groups (called strata) according to some characteristic. Each of these strata is then sampled using one of the other sampling techniques. Cluster Sampling Sampling in which the population is divided into groups (usually geographically). Some of these groups are randomly selected, and then all of the elements in those groups are selected.
Statistics: Introduction Population vs Sample The population includes all objects of interest whereas the sample is only a portion of the population. Parameters are associated with populations and statistics with samples. Parameters are usually denoted using Greek letters (mu, sigma) while statistics are usually denoted using Roman letters (x, s). There are several reasons why we don't work with populations. They are usually large, and it is often impossible to get data for every object we're studying. Sampling does not usually occur without cost, and the more items surveyed, the larger the cost. We compute statistics, and use them to estimate parameters. The computation is the first part of the statistics course (Descriptive Statistics) and the estimation is the second part (Inferential Statistics)
Discrete vs Continuous Discrete variables are usually obtained by counting. There are a finite or countable number of choices available with discrete data. You can't have 2.63 people in the room. Continuous variables are usually obtained by measuring. Length, weight, and time are all examples of continous variables. Since continuous variables are real numbers, we usually round them. This implies a boundary depending on the number of decimal places. For example: 64 is really anything 63.5 <= x < 64.5. Likewise, if there are two decimal places, then 64.03 is really
Print document
In order to print this document from Scribd, you'll anything 63.025 <= x < 63.035. Boundaries always have one more decimal place than the data first need to download it. and end in a 5. Cancel Levels of Measurement
Download And Print
There are four levels of measurement: Nominal, Ordinal, Interval, and Ratio. These go from lowest level to highest level. Data is classified according to the highest level which it fits. Each additional level adds something the previous level didn't have.
Nominal is the lowest level. Only names are meaningful here. Ordinal adds an order to the names. Interval adds meaningful differences Ratio adds a zero so that ratios are meaningful.
Types of Sampling There are five types of sampling: Random, Systematic, Convenience, Cluster, and Stratified.
Random sampling is analogous to putting everyone's name into a hat and drawing out several names. Each element in the population has an equal chance of occuring. While this is the preferred way of sampling, it is often difficult to do. It requires that a complete list of every element in the population be obtained. Computer generated lists are often used with random sampling. You can generate random numbers using the TI82 calculator. Systematic sampling is easier to do than random sampling. In systematic sampling, the list of elements is "counted off". That is, every k th element is taken. This is similar to lining everyone up and numbering off "1,2,3,4; 1,2,3,4; etc". When done numbering, all people numbered 4 would be used. Convenience sampling is very easy to do, but it's probably the worst technique to use. In convenience sampling, readily available data is used. That is, the first people the surveyor runs into. Cluster sampling is accomplished by dividing the population into groups -- usually geographically. These groups are called clusters or blocks. The clusters are randomly selected, and each element in the selected clusters are used. Stratified sampling also divides the population into groups called strata. However, this time it is by some characteristic, not geographically. For instance, the population might be separated into males and females. A sample is taken from each of these strata using either random, systematic, or convenience sampling.
Statistics: Frequency Distributions & Graphs
Print document
Definitions
In order to print this document from Scribd, you'll first need to download it.
Raw Data Cancel Download And Print Data collected in original form. Frequency The number of times a certain value or class of values occurs. Frequency Distribution The organization of raw data in table form with classes and frequencies. Categorical Frequency Distribution A frequency distribution in which the data is only nominal or ordinal. Ungrouped Frequency Distribution A frequency distribution of numerical data. The raw data is not grouped. Grouped Frequency Distribution A frequency distribution where several numbers are grouped into one class. Class Limits Separate one class in a grouped frequency distribution from another. The limits could actually appear in the data and have gaps between the upper limit of one class and the lower limit of the next. Class Boundaries Separate one class in a grouped frequency distribution from another. The boundaries have one more decimal place than the raw data and therefore do not appear in the data. There is no gap between the upper boundary of one class and the lower boundary of the next class. The lower class boundary is found by subtracting 0.5 units from the lower class limit and the upper class boundary is found by adding 0.5 units to the upper class limit. Class Width The difference between the upper and lower boundaries of any class. The class width is also the difference between the lower limits of two consecutive classes or the upper limits of two consecutive classes. It is not the difference between the upper and lower limits of the same class. Class Mark (Midpoint) The number in the middle of the class. It is found by adding the upper and lower limits and dividing by two. It can also be found by adding the upper and lower boundaries and dividing by two. Cumulative Frequency The number of values less than the upper class boundary for the current class. This is a running total of the frequencies. Relative Frequency The frequency divided by the total frequency. This gives the percent of values falling in that class. Cumulative Relative Frequency (Relative Cumulative Frequency) The running total of the relative frequencies or the cumulative frequency divided by the total frequency. Gives the percent of the values which are less than the upper class boundary. Histogram
Print document
In order to print this document from Scribd, you'll A graph which displays the data by using vertical bars of various heights to represent first need to download frequencies. The horizontal axis can be it. either the class boundaries, the class marks, or the class limits. Frequency Polygon Cancel Download And Print A line graph. The frequency is placed along the vertical axis and the class midpoints are placed along the horizontal axis. These points are connected with lines. Ogive A frequency polygon of the cumulative frequency or the relative cumulative frequency. The vertical axis the cumulative frequency or relative cumulative frequency. The horizontal axis is the class boundaries. The graph always starts at zero at the lowest class boundary and will end up at the total frequency (for a cumulative frequency) or 1.00 (for a relative cumulative frequency). Pareto Chart A bar graph for qualitative data with the bars arranged according to frequency. Pie Chart Graphical depiction of data as slices of a pie. The frequency determines the size of the slice. The number of degrees in any slice is the relative frequency times 360 degrees. Pictograph A graph that uses pictures to represent data. Stem and Leaf Plot A data plot which uses part of the data value as the stem and the rest of the data value (the leaf) to form groups or classes. This is very useful for sorting data quickly.
Statistics: Grouped Frequency Distributions
Guidelines for classes 1. There should be between 5 and 20 classes. 2. The class width should be an odd number. This will guarantee that the class midpoints are integers instead of decimals. 3. The classes must be mutually exclusive. This means that no data value can fall into two different classes 4. The classes must be all inclusive or exhaustive. This means that all data values must be included. 5. The classes must be continuous. There are no gaps in a frequency distribution. Classes that have no values in them must be included (unless it's the first or last class which are dropped). 6. The classes must be equal in width. The exception here is the first or last class. It is possible to have an "below ..." or "... and above" class. This is often used with ages.
Creating a Grouped Frequency Distribution 1. Find the largest and smallest values 2. Compute the Range = Maximum - Minimum 3. Select the number of classes desired. This is usually between 5 and 20.
Print document
In dividing order to print this document from Scribd, you'll and rounding up. There are 4. Find the class width by the range by the number of classes firstof need to download two things to be careful here. You must it. round up, not off. Normally 3.2 would round to be 3, but in rounding up, it becomes 4. If the range divided by the number of classes gives an integer value (no remainder), then you can either add one to the number of classes or add one to the Cancel Download And Print class width. Sometimes you're locked into a certain number of classes because of the instructions. The Bluman text fails to mention the case when there is no remainder. 5. Pick a suitable starting point less than or equal to the minimum value. You will be able to cover: "the class width times the number of classes" values. You need to cover one more value than the range. Follow this rule and you'll be okay: The starting point plus the number of classes times the class width must be greater than the maximum value . Your starting point is the lower limit of the first class. Continue to add the class width to this lower limit to get the rest of the lower limits. 6. To find the upper limit of the first class, subtract one from the lower limit of t he second class. Then continue to add the class width to this upper limit to find the rest of the upper limits. 7. Find the boundaries by subtracting 0.5 units from the lower limits and adding 0.5 units from the upper limits. The boundaries are also half-way between the upper limit of one class and the lower limit of the next class. Depending on what you're trying to accomplish, it may not be necessary to find the boundaries. 8. Tally the data. 9. Find the frequencies. 10. Find the cumulative frequencies. Depending on what you're trying to accomplish, it m ay not be necessary to find the cumulative frequencies. 11. If necessary, find the relative frequencies and/or r elative cumulative frequencies.
12.
Statistics: Data Description
13. 14. Definitions 15. Statistic 16. Characteristic or measure obtained from a sample 17. Parameter 18. Characteristic or measure obtained from a population 19. Mean 20. Sum of all the values divided by the number of values. This can either be a population mean (denoted by mu) or a sample mean (denoted by x bar) 21. Median 22. The midpoint of the data after being ranked (sorted in ascending order). There are as many numbers below the median as above the median. 23. Mode 24. The most frequent number 25. Skewed Distribution 26. The majority of the values lie together on one side with a very few values (the tail) to the other side. In a positively skewed distribution, the tail is to the right and the mean is larger than the median. In a negatively skewed distribution, the tail is to the left and the mean is smaller than the median. 27. Symmetric Distribution 28. The data values are evenly distributed on both sides of the mean. In a symmetric distribution, the mean is the median.
Print document
In order to print this document from Scribd, you'll 29. Weighted Mean first needis to multiplied download it.by its weight and summed. This sum is divided 30. The mean when each value by the total of the weights. 31. Midrange Cancel Download And Print 32. The mean of the highest and lowest values. (Max + Min) / 2 33. Range 34. The difference between the highest and lowest values. Max - Min 35. Population Variance 36. The average of the squares of the distances from the population mean. It is the sum of the squares of the deviations from the mean divided by the population size. The units on the variance are the units of the population squared. 37. Sample Variance 38. Unbiased estimator of a population variance. Instead of dividing by the population size, the sum of the squares of the deviations from the sample mean is divided by one less than the sample size. The units on the variance are the units of the population squared. 39. Standard Deviation 40. The square root of the variance. The population standard deviation is the square root of the population variance and the sample standard deviation is the square root of the sample variance. The sample standard deviation is not the unbiased estimator for the population standard deviation. The units on the standard deviation is the same as the units of the population/sample. 41. Coefficient of Variation 42. Standard deviation divided by the mean, expressed as a percentage. We won't work with the Coefficient of Variation in this course. 43. Chebyshev's Theorem 44. The proportion of the values that fall within k standard deviations of the mean is at least
where k > 1. Chebyshev's theorem can be applied to any distribution regardless of its shape. 45. Empirical or Normal Rule 46. Only valid when a distribution in bell-shaped (normal). Approximately 68% lies within 1 standard deviation of the mean; 95% within 2 standard deviations; and 99.7% within 3 standard deviations of the mean. 47. Standard Score or Z-Score 48. The value obtained by subtracting the mean and dividing by the standard deviation. When all values are transformed to their standard scores, the new mean (for Z) will be zero and the standard deviation will be one. 49. Percentile 50. The percent of the population which lies below that value. The data must be ranked to find percentiles. 51. Quartile 52. Either the 25th, 50th, or 75th percentiles. The 50th percentile is also called the median. 53. Decile 54. Either the 10th, 20th, 30th, 40th, 50th, 60th, 70th, 80th, or 90th percentiles. 55. Lower Hinge
Print document
In order to print this document from Scribd, you'll 56. The median of the lower half of the numbers (up to and including the median). The lower first need to download it. hinge is the first Quartile unless the remainder when dividing the sample size by four is 3. 57. Upper Hinge 58. The median of the upper half of the numbers (including the median). The upper hinge is Cancel Download And Print the 3rd Quartile unless the remainder when dividing the sample size by four is 3. 59. Box and Whiskers Plot (Box Plot) 60. A graphical representation of the minimum value, lower hinge, median, upper hinge, and maximum. Some textbooks, and the TI-82 calculator, define the five values as the minimum, first Quartile, median, third Quartile, and maximum. 61. Five Number Summary 62. Minimum value, lower hinge, median, upper hinge, and maximum. 63. InterQuartile Range (IQR) 64. The difference between the 3rd and 1st Quartiles. 65. Outlier 66. An extremely high or low value when compared to the rest of the values. 67. Mild Outliers 68. Values which lie between 1.5 and 3.0 times the InterQuartile Range below the 1st Quartile or above the 3rd Quartile. Note, some texts use hinges instead of Quartiles. 69. Extreme Outliers 70. Values which lie more than 3.0 times the InterQuartile Range below the 1st Quartile or above the 3rd Quartile. Note, some texts use hinges instead of Quartiles
71.
Stats: Probability
72. 73.
Definitions
74. Probability Experiment 75. Process which leads to well-defined results call outcomes 76. Outcome 77. The result of a single trial of a probability experiment 78. Sample Space 79. Set of all possible outcomes of a probability experiment 80. Event 81. One or more outcomes of a probability experiment 82. Classical Probability 83. Uses the sample space to determine the numerical probability that an event will happen. Also called theoretical probability. 84. Equally Likely Events 85. Events which have the same probability of occurring. 86. Complement of an Event 87. All the events in the sample space except the given events. 88. Empirical Probability 89. Uses a frequency distribution to determine the numerical probability. An empirical probability is a relative frequency. 90. Subjective Probability
Print document
In order to print this document from Scribd, you'll 91. Uses probability values based on an educated guess or estimate. It employs opinions and first need to download it. inexact information. 92. Mutually Exclusive Events 93. Two events which cannot happen at theDownload same time.And Print Cancel 94. Disjoint Events 95. Another name for mutually exclusive events. 96. Independent Events 97. Two events are independent if the occurrence of one does not affect the probability of the other occurring. 98. Dependent Events 99. Two events are dependent if the first event affects the outcome or occurrence of the second event in a way the probability is changed. 100. Conditional Probability 101. The probability of an event occurring given that another event has already occurred. 102. Bayes' Theorem 103. A formula which allows one to find the probability that an event occurred as the result of a particular previous event.
