Statistics vs. statistics
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1 Statistics vs. statistics Question: What is Statistics (with a capital S)? Definition: Statistics is the science of collecting, organizing, summarizing and interpreting data. Note: There are 2 main ways to summarize data 1. With a picture or a graph 2. With a number or small collection of numbers that tell you something about the data
2 Statistics vs. statistics Note: There are 2 main ways to summarize data 1. With a picture or a graph 2. With a number or small collection of numbers that tell you something about the data Question: What is a statistic (with a lowercase s)? Definition: A statistic is a number that can be calculated from a set of data that summarizes the data in some way
3 Example of statistics Ex: The following data gives the ages of the students in one of my previous statistics classes. Student Ages Come up with some statistics for this data set.
4 Chapter 3: Numerically Summarizing Data Sec. 3.1: Measures of Central Tendency (measures of center) Sec. 3.2: Measures of Dispersion (measures of spread)
5 Sec. 3.1: Measures of Central Tendency (measures of center)
6 Sec. 3.1: Measures of Central Tendency (measures of center) Why would you want to measure the center of data? Story
7 Sec. 3.1: Measures of Central Tendency (measures of center) There are 4 ways of measuring the center 1) Name: MEAN (or AVERAGE) Symbol: xҧ How to calculate: Add all of the data points and divide by the total number of data points
8 Sec. 3.1: Measures of Central Tendency (measures of center) There are 4 ways of measuring the center 2) Name: MEDIAN Symbol: x How to calculate: 1. Put the data in increasing order 2. If n is odd, the median is the middle number 3. If n is even, the median is the average of the 2 middle numbers
9 Sec. 3.1: Measures of Central Tendency (measures of center) There are 4 ways of measuring the center 3) Name: MODE How to calculate: The number that appears the most often There can be more than one mode If no number repeats, there is no mode
10 Sec. 3.1: Measures of Central Tendency (measures of center) There are 4 ways of measuring the center 4) Name: MIDRANGE How to calculate: The average of the lowest and highest data points (i.e. add the lowest and highest numbers and divide by 2)
11 Sec. 3.1: Measures of Central Tendency (measures of center) Ex 1: Find the mean, median, mode and midrange of the following data set Data: 12, 4, 15, 12, 10, 10, 12, 2, 19, 10, 4
12 Sec. 3.1: Measures of Central Tendency (measures of center) Ex 2: Find the mean, median, mode and midrange of the following data set Data: 23, 17, 8, 19, 4, 11, 31, 2
13 Sec. 3.2: Measures of Dispersion (measures of spread)
14 Sec. 3.2: Measures of Dispersion (measures of spread) Why would you want to measure the spread of data? Story
15 Sec. 3.2: Measures of Dispersion (measures of spread) There are 3 ways of measuring the spread 1) Name: RANGE How to calculate: Highest data point lowest data point
16 Sec. 3.2: Measures of Dispersion There are 3 ways of measuring the spread 2) Name: STANDARD DEVIATION Symbol: s (measures of spread) How to calculate: The square root of the almost average of the squares of the distance each data point is from the mean of the entire data set
17 Sec. 3.2: Measures of Dispersion (measures of spread) There are 3 ways of measuring the spread 3) Name: VARIANCE Symbol: s 2 How to calculate: The square of the standard deviation
18 Sec. 3.2: Measures of Dispersion (measures of spread) Ex 3: Derive the formula for the standard deviation s of a data set by finding the standard deviation of the data set below Data: 1, 2, 4, 11, 12
19 Formulas If N stands for the population size and n stands for the sample size, then... Population Mean μ = σ x N Sample Mean x ҧ = σ x n
20 Formulas If N stands for the population size and n stands for the sample size, then... Population Standard Deviation σ = σ x μ 2 N Sample Standard Deviation s = σ x xҧ 2 n 1 = σ x 2 σ x 2 n n 1
21 Formulas If N stands for the population size and n stands for the sample size, then... Population Variance σ 2 = σ x μ 2 N Sample Variance s 2 = σ x xҧ 2 n 1 = σ x 2 σ x 2 n n 1
22 Sec. 3.2: Measures of Dispersion (measures of spread) Ex 4: Find the range, standard deviation, and variance of the following data set Data: 23, 17, 8, 19, 4, 11, 31, 2
23 Going Between Standard Deviation and Variance Ex 5: If the standard deviation of a data set is 9, what is the variance of the data set? Ex 6: If the variance of a data set is 9, what is the standard deviation of the data set?
24 Some Other Questions Ex 7: Suppose you have 2 data sets. Let s 1 be the standard deviation of data set 1 and let s 2 be the standard deviation of data set 2. If s 1 = 8.3 and s 2 = 4.1, what can you say about the data sets?
25 Some Other Questions Ex 8: Below are 2 data sets. Data set 1: 755, 753, 756, 757, 751 Data set 2: 1, 3, 12, 76, 163 Let s 1 be the standard deviation of data set 1 and let s 2 be the standard deviation of data set 2. Without calculating them, which one is bigger: s 1 or s 2? Why?
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