Sampling, frequency distribution, graphs, measures of central tendency, measures of dispersion

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1 Statistics Basics Sampling, frequency distribution, graphs, measures of central tendency, measures of dispersion Part 1: Sampling, Frequency Distributions, and Graphs The method of collecting, organizing, analyzing, and interpreting data, as well as drawing conclusions based on the data is called statistics. Methodology is divided into two main areas. Descriptive Statistics: Collecting, organizing, summarizing, and presenting data. Inferential Statistics: Making generalizations about and drawing conclusions from the data collected. We have the following basic concepts in statistics: Population: Set containing all the people or objects whose properties are to be described and analyzed by the data collector. Sample: Subset or subgroup of the population. Representative Sample: A sample that exhibits characteristics typical of those possessed by the target population. A random sample is a sample obtained in such a way that every element in the population has an equal chance of being selected. Methodology: Identify each element in the population. Assign numbers to each element in the population. Randomly select numbers. Assign the elements in the population who have those numbers to the sample set. A frequency distribution is an arrangement of the values that one or more variables take in a sample. Each entry in the table contains the frequency or count of the occurrences of values within a particular group or interval, and in this way, the table summarizes the distribution of values in the sample.

2 Ex: If 100 people rate a five-point Likert scale assessing their agreement with a statement on a scale on which 1 denotes strong agreement and 5 strong disagreement, the frequency distribution of their responses might look like: Rank Degree of agreement Number 1 Strongly agree 20 2 Agree somewhat 30 3 Not sure 20 4 Disagree somewhat 15 5 Strongly disagree 15 We can group the frequencies into classes that are meaningful for the data. Ex: The heights of the students in a class could be organized into the following frequency table. Height range Number of students Cumulative number feet feet feet feet The class has 4.5 as the lower class limit and 4.9 as the upper class limit. The class width is 0.5. A frequency distribution shows us a summarized grouping of data divided into mutually exclusive classes and the number of occurrences in a class. It is a way of showing unorganized data e.g. to show results of an election, income of people for a certain region, sales of a product within a certain period, student loan amounts of graduates, etc. A very used tool in statistics is the graph: Histogram: A bar graph with bars that touch can be used to visually display the data. Frequency Polygon: A line graph formed by connecting dots in the midpoint of each bar of the histogram Stem-and-Leaf Plots: This plot is constructed by separating each data item into two parts: Stem consists of the ten s digit and Leaf consists of the units digit. Ex: An example histogram of the heights of 31 Black Cherry trees:

3 Ex: A frequency polygon for a data set containing the scores 1,1,2,2,2,2,2,3,3,3,3,4,4,5: Ex: A steam-and-leaf plot suggesting the ages of people at a family reunion. In this table the steam 3 and leaf 4 means that one people has the age of 34 years.

4 Part 2: Measures of Central Tendency Mean: The sum of the data items divided by the number of items., where Σx represents the sum of all the data items and n represents the number of items. Ex: The arithmetic mean of six values: 34, 27, 45, 55, 22, 34 is When many data values occur more than once and a frequency distribution is used to organize the data, we can use the following formula to calculate the mean:, Where x represents each data value, f represents the frequency of that data value, and Σxf represents the sum of all the products obtained by multiplying each data value by its frequency. Here the number n represents the total frequency of the distribution. Ex: Ages of Statistics Students in Spring 2000: Ages Frequency

5 Estimate the mean for this set of data. Class Midpoint Frequency m f The sum of the product of the midpoints and frequencies is 1005 (add the values in the last column). Divide this number by 40 (the total of the frequencies), and we estimate the mean to be Median is the data item in the middle of each set of ranked, or ordered, data. To find the median of a group of data items: Ex: is 4. Arrange the data items in order, from smallest to largest. If the number of data items is odd, the median is the data item in the middle of the list. If the number of data items is even, the median is the mean of the two middle data items. a. The median for the set {2,2,5,3,2,5,4,7,4}: we have {2,2,2,3,4,4,5,5,7}, so the median b. The median for the set {4,5,2,9,1,4,9,6}: we have {1,2,4,4,5,6,9,9}, so we add the middle terms = 9 and divide by 2: the mean is 4.5. Ex: Finding the Median for a Frequency Distribution of the stress-level rating: Stress Rating x Frequency f

6 There are 151 data items in this table so n = 151. The median is the value in the 76 position. Count down the frequency column in the distribution until we identify the 76 th data item; we find it to be 7. So, the median stress-level rating is 7 The mode is the data value that occurs most often in a data set. If more than one data value has the highest frequency, then each of these data values is a mode. If no data items are repeated, then the data set has no mode. Ex: Find the mode for the following groups of data: 7, 2, 4, 7, 8, 10. The mode is 7. The midrange is found by adding the lowest and highest data values and dividing the sum by 2:. 2 Ex: The midrange for the data set {10, 23, 8, 9, 14, 11} is Part 3: Measures of Dispersion The range is used to describe the spread of data items in a data set. Two of the most common measures of dispersion are range and standard deviation. It is given by the difference between the highest and the lowest data values in a data set:

7 Range = highest data value lowest data value Ex: Honolulu s hottest day is 89º and its coldest day is 61º. The range in temperature is: 89º 61º = 28º. Ex: Find the deviations from the mean for 472 in the five data items 778, 472, 147, 106, and 82. First, the mean is = Deviation from mean = data item mean = Standard deviation is a used measurement of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average (mean, or expected value). A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values. Computing the standard deviation for a data set: Find the mean; Find the deviation of each data item from the mean: (data item mean) Square each deviation: data item mean ; Add the squared deviations: data item mean ; Divide:, where n is the number of data items; Take the square root:. Ex: Find the standard deviation for the data set: 778, 472, 147, 106, and 82. The mean was found to be 317. Deviations: =461, =155, =-170, =-211, =-235. Square the deviations and add them: ,192. Because there are 5 items, we divide the previous number by 5-1=4 and obtain:

8 365,192 91, Finally, we apply the square root: 91, The standard deviation is approximately