1 Overview of Statistics; Essential Vocabulary

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1 1 Overview of Statistics; Essential Vocabulary Statistics: the science of collecting, organizing, analyzing, and interpreting data in order to make decisions Population and sample Population: the entire set of individuals of interest The population is determined by a problem. Sample: a subset of a population i.e., members of the population which you actually know something about Two branches of statistics Descriptive statistics: organizing, summarizing, and displaying data Inferential statistics: use of a sample to draw conclusions about a population To make an inference means to draw a general conclusion about a population from a sample. Probability is importantly involved in inferential statistics and not at all involved in descriptive statistics. Variable, Data, Parameter, Statistic Variable: a characteristic of an individual, to be measured or observed E.g., if the individuals are people, variables might be height, eye color,... Data: information from counting, measuring, or observing I.e., what you write down about individuals in your sample or population. Note that data point, data value, and data entry all mean the same thing. Parameter: a value computed from population data Statistic: a value computed from sample data A statistic is an estimate of a parameter. In real life, you hardly ever have a parameter. But we re interested in parameters (on average, how much does a baby weigh at birth?), and we use statistics to estimate them, so we need words to distinguish the two. E.g. Jake measured the diameters of 100 ball bearings chosen from a shipment of 10,000. The average diameter was 1.1mm. Population: Sample: Variable: Data: Statistic: A statistic changes when the sample changes. E.g. A 2009 survey of 218 law firms with at least 50 lawyers found that 69% of firms had cut personnel in the previous year. (Altman-Weil). Is the 69% a parameter or a statistic? 1

2 Univariate and bivariate data Data is univariate if is made up of a list of individual values. Paired or bivariate data is made up of ordered pairs of numbers (often presented in a table). E.g. John measured the boiling point of water at various altitudes. Altitude (ft) Boiling point ( F) As ordered pairs (0, 212) (1000, 210.2) (2000, 208.4) (5280, ) Experiment Another basic term: An experiment is any activity with measurable outcomes, e.g., rolling a die, drawing a card, or medicating patients. Each possible result of an experiment is called an outcome. 2

3 Hey, girls: here s what my grandfather s sister was up to in 1931: That s the Proceedings of the National Academy of Sciences, ladies. Get your Margaret Hilferty on! 3

4 2 Qual vs. Quant; Discrete vs. Continuous; Sampling Qualitative and quantitative data Two basic types of data: quantitative (or numerical): numbers that are the results of measuring or counting It makes sense to do arithmetic on quantitative data. It usually makes sense to average it. qualitative (or categorical): everything else It does not make sense to do arithmetic on qualitative data. E.g. Qual or quant? (a) diameters of Eastern White Pines (b) eye color (c) numbers on jerseys of starting team Discrete and continuous data We say that a particular set of numerical data is continuous if data values can be any number in an interval, and discrete if they can only be one of a set of numbers that can be counted out. To figure out whether data is discrete or continuous, think of plotting all possible values on a number line. If there would be gaps between values, it s discrete; otherwise, it s continuous. E.g. Discrete or continuous? What are the possible values? (a) Number of (whole, unbroken) eggs in one carton (b) Weight of one carton of eggs, in ounces (c) Time since the last customer arrived at Joe s diner, in seconds (d) Number of stocks in the DJIA the share prices of which closed higher than they opened yesterday We ll return to Discrete vs. Continuous later. 4

5 Sampling Census: use the entire population (rarely possible) We want to be able to use information about a sample to infer something about a population characteristic, so we want a sample that is representative of the population. A sampling method is biased if it tends to produce samples that are not representative of the population. Sometimes we refer to such samples as biased samples. We say that a sample is not representative if statistics computed from it don t correctly estimate the parameters they re supposed to estimate. Sampling error: difference between the true value of a parameter and the value of the statistic that estimates that parameter A simple random sample is one in which every possible sample of the same size has the same chance of being selected. This is not quite the same thing as saying that every individual has the same chance of being selected. E.g., a coin is flipped. If it comes up heads, Alice and Bob are both chosen; if it comes up tails, Carol is chosen. Then each individual has a 50/50 chance of being chosen, but Bob and Carol cannot both be chosen. You get a simple random sample by assigning a number to every member of the population and then using a random number generator to choose. A simple random sample is almost always best, but sometimes you cannot afford a large enough one. We will assume that all samples are simple random samples unless told otherwise. 5

6 3 Frequency and Relative Frequency Distributions; Histograms; Common Distribution Shapes Frequency distributions A frequency distribution for a collection of numbers is a table with two columns. In the simplest kind of frequency table, the first column contains each number that occurs, and the second column contains the number of times the number occurs (its frequency). E.g. Make a frequency distribution table for the numbers 3, 4, 5, 3, 6, 2, 7, 6. Point Frequency This sort of frequency distribution, in which each point is listed, is said to be ungrouped. For a large collection of numbers, a grouped frequency distribution is often useful. These have numbers grouped into intervals or classes. The frequency of each class is the number of numbers in it. E.g. The attendance in an Intro Stat section on each day of one semester was as follows: 45, 47, 43, 40, 38, 36, 23, 35, 44, 26, 32, 35, 40, 38, 38, 39, 36, 37, 45, 35, 36, 37, 38, 36, 33, 35, 36, 40, 45 To make a frequency distribution for this data using the classes 20 30, 30 40, 40 50, we just count the number of numbers in each class: Class # days Class Frequency In a frequency distribution, an interval specified by a dash between two numbers, like 27 31, is considered to contain its left endpoint but not its right endpoint. E.g., contains 27 but not 31. In interval notation, this would be [27,31). In a frequency distribution for grouped data, the class width is the left-hand endpoint of any interval minus the left-hand endpoint of the next lower interval. E.g. For the example above, the class width is = 10. Recall that the midpoint of an interval with endpoints a and b is a+b 2. 6

7 Frequency histograms A frequency histogram is a specialized bar graph of a frequency distribution table. Horizontal axis: classes bars must touch label with class boundaries or midpoints Vertical axis: frequencies E.g. Construct a frequency histogram for the table shown. Interval Freq Relative frequency distributions The relative frequency of a class is # data points in the class # data points in the whole data set Relative frequencies can be given as decimals or as percentages. A relative frequency distribution table shows percentages of the whole for each class instead of the number in each class. Note that the percentages must add up to 100. We make a relative frequency distribution from a frequency distribution. E.g. Compute the relative frequencies for the attendance data. Class Frequency Relative Frequency (%)

8 Relative frequency histograms A relative frequency histogram is a histogram for a relative frequency distribution. E.g. Make a relative frequency histogram for the frequency distribution shown. The Greek capital letter sigma (Σ) means add up the values (i.e., sum ). Class Frequency Relative Frequency (%) Σ f = 29 Common distribution shapes Memorize: Note that (i) distributions don t have to be exact and (ii) lots of distributions are symmetrical, but only the one that is symmetrical and has a single central hump is called the symmetric distribution. 8

9 4 Cumulative Frequency; Other Pictures of Data Cumulative frequency The cumulative frequency at any row of a frequency table is the sum of all frequencies up to and including that row. We make a cumulative frequency distribution from a frequency distribution. E.g. Add a column for cumulative frequency to the frequency distribution. Class Frequency Cumulative Frequency E.g. Make a cumulative frequency graph for the table above. We first plot points (x,y) where x is the endpoint of an interval, and y is the cumulative frequency at that interval. Then we join the points into a line graph. Pie charts Pie charts are mostly used for categorical data. For univariate data 9

10 Bar graphs Pretty simple. The bars should be separated. Bar charts are used for both categorical and numerical data. Pareto charts A Pareto chart is just a bar chart for categorical data in which the categories are ordered from highest to lowest frequency as you go from left to right. 10

11 Line graphs For bivariate data (ordered pairs). Put a dot at the point represented by each ordered pair and then connect the dots with straight line segments. Time series charts A line graph of a quantity or quantities at regularly spaced times. (The one below is a double line graph.) 11

12 Scatter plots For bivariate, numerical data (ordered pairs of numbers) Just graph the ordered pairs Very useful for seeing patterns in data We ll mostly use scatter plots later in the course. 12

13 5 Mean, Median, Mode; Outliers A measure of central tendency is an average: a single value intended to be typical of the data set. It s a value intended answer the question Where is the middle of most of the data? These are for univariate data only For a numerical data set, a measure of central tendency tells you about where the middle of the data lies on the number line. Notation: Number of data points in a population: N Number of data points in a sample: n Mean For quantitative data only If the data are x 1,x 2,...,x k, then the mean is x 1+ +x k k Notation: population mean: µ sample mean: x The mean has the same units as the data. (for population or sample). E.g. Find the mean of the following data. 52, 52, 54, 57, 58, 63, 63, 65, 67, 67, 70, 71, 72, 73, 75, 76, 77, 93 Think of x as an estimate of µ. For any one sample, it might happen that x is far from µ, but it can be proved that if many samples are taken from the same population, then we can expect the average value of x to be very close to µ. This means that x is an unbiased estimator for µ. Median for quantitative data only The median Q 2 is the number halfway up a sorted list of data. You will need to know that the middle position in a list with k entries is the k+1 2 position. (This is not the same as the middle entry.) To find the median of a dataset with k values: 1. Sort the data in ascending order. 2. If k is odd, then the median is the middle data value. If k is even, then the median is the mean of the two middle values. E.g. Find the median of the data set 2, 7, 3, 4, 8. 13

14 E.g. Find the median of the following data. 52, 52, 54, 57, 58, 63, 63, 65, 67, 67, 70, 71, 72, 73, 75, 76, 77, 93 Either the mean or the median may be called an average. Mode The mode is most useful for quantitative data, but there are times when you must give a typical value for a qualitative data set. E.g. A car dealership sold 60 cars in the past week of which 40 were red, 12 were green, and 8 were blue. If forced to describe the average color of car sold using one or two values or the phrase no typical value, what response would you give? What if 25 were red, 25 were green, and 10 were blue? What if 20 were red, 20 were green, and 20 were blue? If there is a data value that occurs most frequently, it is called the mode of the data. If there are two values that occur most frequently, the data is bimodal and we report both values. If there are more than two values that occur most frequently, the data is multimodal, and we report no mode. E.g. Find the mode(s) of each data set. (a) a, a, b, c, c, c, d, d, e, f, g (b) a, b, c, d, d, e, e, f, g (c) a, b, c, c, d, d, e, f, f, g, g, h (d) a, b, c, d, e Bimodal data is often a sign that your measurements are really from two populations. 14

15 Outliers There s a problem with data points that are far away from most of the data. E.g. Annual compensation for ten RU employees with faculty rank (in $): 47,561; 49,687; 52,375; 53,626; 60,573; 63,716; 73,832; 96,666; 105,719; 508,299. The mean of this data is $111,205.40; the median is $62, Which average is most representative of the center of this data? (If you were recruiting a prospective faculty member, which would you feel most honest reporting?) A data value is a (suspected) outlier if it is extremely high or low compared to the rest of the data. A histogram for a data set with an outlier often looks like the one below. Our only method for finding outliers is to look at a data plot or histogram. There are numerical methods, but we won t study them in this course. How to choose the measure of central tendency to use As the example showed, the mean is strongly affected by outliers, but the median isn t. Therefore, we follow these rules to choose the measure of central tendency to use: If the data set contains qualitative data, use the mode. If there is an outlier (or two) in a set of data, use the median. Use the mean in all other situations. 15

16 Measures of central tendency on the calculator To find measures of central tendency on the TI, you must put the data in list L1 and then use the 1-Var Stats function. There s more on 1-Var Stats in Lecture 6. 16

17 6 Range, Variance, Standard Deviation; Weighted Mean Range, variance, std dev are all measures of dispersion = spread = variability = how spread out the data is Another measure of dispersion that we will study later is the interquartile range. These measures of dispersion are for univariate, quantitative data only. E.g. The example data sets below all have the same mean and median, namely, 7. It s the spread or variability that s different the range of the data points and how far they tend to be from their center. Data Set A: 3, 5, 6.4, 6.7, 6.8, 7, 7.3, 7.4, 7.7, 8.7, 11 Data Set B: 1.25, 1.3, 1.35, 1.4, 1.4, 1.7, 3.1, 4, 7, 7, 9.7, 12, 12.05, 12.35, 12.4, 12.55, 12.7, Data Set C: 16.9, 16.5, 15.9, 15.8, 15.75, 15.7, 15.7, 15.5, 12, 6.65,6.8,6.9,7,7,7.2,7.4,7.8,27,29.4,29.6,29.7,29.75,29.75,29.9,29.9,30 Range Range = (max data value) (min data value) computed the same way for population & sample data E.g. range of A : 11 3 = 8 range of B : = 11.5 range of C : 30 ( 16.9) = 46.9 Note that the range is very sensitive to outliers. Variance The variance will measure how far the data tends to be from its mean. We will define the sample variance s 2 and then the population variance σ 2. We need to do a little work before we can define either. If x is a data point in a sample, then the deviation of x (from the mean) is x x. E.g. For data set A, x = 7. Thus the deviation of the data point 3 from the mean is 3 7 = 4. The deviation measures how far x is from the mean. You might think that we could use the average deviation to measure how far the data tends to be from its mean, but there s a problem. Problem: the sum of all the deviations is always 0. Solution: use the squares of the deviations to measure how far data points are from x. 17

18 The sample variance is s 2 = sum of (deviations squared) number of data points 1 (x x)2 = n 1 The denominator is n 1 because it can be shown that if n is used, then for samples that are small compared to the size of the population, the result is a biased estimator for σ 2, while using n 1 makes it unbiased. Again, unbiased means that if many samples are taken and s 2 is computed for each one, then we can expect the average value of s 2 to be very close to the population variance. E.g. Find the variance of the sample 1.5, 1.7, 1.9, 2.1, 2.3. x = x x x (x x) (x x) 2 = s 2 = The variances of data sets A, B, and C are (approximately) 4, 25, and 365, respectively. The population variance is σ 2 = Σ(x µ)2 N. The units of the variance are the units of the data squared. Larger variance corresponds beautifully to greater variability, but the units are wrong. E.g., if the data points represent the number of shoes in a man s closet, then the units of the variance are shoes squared. We can fix this. Standard deviation The standard deviation is the positive square root of the variance: For population data: σ = σ 2 For sample data: s = s 2 E.g. The standard deviations of data sets A, B, and C are (approximately) 2.0, 5.0, and 19.1, respectively. The units of the std dev are the units of the data. Interpret std dev as giving the average distance of a data point from the mean. 18

19 E.g. The table below shows the numbers of pairs of shoes in four men s closets. Find the mean, median, range, and standard deviation. Interpret the standard deviation. Person Pairs of shoes A 12 B 4 C 8 D 7 x = # pairs of shoes mean: median: range: x x x (x x) (x x) 2 = s = E.g. John recorded the weights of 300 randomly chosen apples, while Jane recorded the weights of 300 randomly chosen dogs. Which data set had the higher standard deviation? Range, variance, standard deviation on the calculator The TI s 1-Var Stats function, introduced in the previous section, gives you much more than just the mean and median. It has two pages of output; you must arrow down to get to the second page. Remember that the calculator does not know whether you entered population or sample data, so it shows you values computed from both when they re computed differently. Recall that the midpoint of the interval [a,b] is a+b 2. (You ll need this in the exercises.) 19

20 Weighted mean For quantitative data only Sometimes we have data from categories some of which are more important than others. E.g., when computing a GPA, As are weighted more heavily than Bs. In these cases, we must compute a weighted mean. If there are k categories, the weights of the categories are w 1,w 2,...,w k, and the data values in those categories are x 1,x 2,...,x k, then the weighted mean is Note: Some authors use x to denote weighted means. x w = x 1w 1 + x 2 w x k w k w 1 + w w k E.g. Jasmine has 22 credit hours of courses in which she received an A, 26 in which she received a B, 12 in which she received a C, and 6 in which she received a D. Assuming the usual scale (A = 4, B = 3, etc.), what is her GPA? Chebyshev s Theorem Chebyshev s Theorem connects the standard deviation, which is computed from the data, to the probability that a certain amount of the data is within a given distance of the mean. Chebyshev s Theorem: No matter how the data are distributed, the portion of the data lying within k standard deviations of the mean, for k > 1, is at least 1 1 k 2. k 1 1 k = = 75% so at least 75% of the data lies within two std devs of the mean % % lies within three std devs of the mean = 93.75% = 96% 96% The mean tells us about where the middle of the data is. C s Thm says that the standard deviation tells us how likely we are to find a data point near the mean. Illustrations of Chebyshev s Theorem for a few data sets Chebyshev s Theorem tells us that the standard deviation can be useful for estimating probabilities. For example, it says that if a data value is chosen at random from any data set, then the probability that the chosen value is within two standard deviations of the mean is at least 75%. Chebyshev s Theorem is actually very conservative. When we know more about a distribution, we can usually get much better estimates. 20

21 7 Quantiles; Five-Number Summary; Box Plots; IQR; z-score Quantiles (or fractiles) are numbers that split an ordered list of numbers into parts each with approximately the same number of data points. The simplest is the median, which splits an ordered list into two parts. Quartiles About 1/4 of the data points are less than Q 1 1/2 Q 2 = median 3/4 Q 3 There are several ways of computing the various fractiles; we will use the calculator. E.g. Find Q 1,Q 2, and Q 3 for the data set Percentiles We won t compute these it s rather complicated but we ll learn to interpret them. Percentiles split an ordered list of numbers into 100 approximately equal parts. Notation: P 1,P 2,...,P 99 About k percent of the data is to the left of P k. Equivalently, if a data point is chosen at random, then the probability that it is less than P k is about k 100. The percentile rank of a data point doesn t tell us anything about the data point s size. It tells us about its rank how far up the list it is. 21

22 E.g. Consider the cumulative frequency graph of SAT scores at a particular school. (a) What test score corresponds to the 70th percentile? (b) About what percentage of all test-takers got a score higher than 1200? (c) If a test-taker is chosen at random, what s the probability that his score is less than 1200? A point (x,y) is on this graph if y% of test-takers got a score of at least x. Five-number summary of a data set (min, Q 1, Q 2, Q 3, max) E.g. Construct the 5-number summary for the data set

23 Box plots (boxplots, box-and-whisker plots) Just a graph of a 5-number summary: E.g. Construct the box plot for the data in the previous example. Interquartile range The interquartile range or IQR of a data set is the difference between the first and third quartiles: IQR = Q 3 Q 1 The IQR is a measure of dispersion (spread). z-score The z-score or (standard score) of a data point x measures how far x is from the mean in units of the standard deviation. That is, it measures how far the data point is from being average. For a sample, the formula is z = two decimal places. z = x µ σ The z-score formula x x s. With two exceptions that we will see later, z-scores should always be rounded to To understand the formula, note that the signed distance from x to µ on the number line is x µ and that the number of lengths σ in a length x µ is (x µ) /σ. E.g. A data set has mean 17.5 and standard deviation 6. What is the z-score of the data value 21.2? 23

24 Index Symbols x, 14 1-Var Stats, 20 B bar graph, 11 bimodal, 15 bivariate, 3 box plot, 1 C categorical data, 5 Chart Pareto, 11 pie, 10 time series, 12 Chebyshev s Theorem, 21 classes, in frequency distributions, 7 continuous data, 5 Cumulative frequency, 10 D Data, 2 bivariate, 3 categorical, 5 continuous, 5 discrete, 5 grouped, 7 numerical, 5 numerical data, 5 qualtitative, 5 quantitative, 5 ungrouped, 7 univariate, 3 deviation, 18 discrete data, 5 dispersion, 18 Distribution common shapes, 9 frequency, 7 relative frequency, 8 E Error sampling, 6 Estimator unbiased, 14 Experiment, 3 F Five-number summary, 23 Frequency, 7 cumulative, 10 distribution, 7 histogram, 8 relative, 8 G Graph bar, 11 line, 12 scatter plot, 13 grouped data, 7 H Histogram frequency, 8 relative frequency, 9 I Interquartile range, 1 IQR, 1 L line graph, 12 M Mean, 14 arithmetic, 21 weighted, 21 Measure of central tendency, 14 Median, 14 midpoint, 20 Mode, 15 multimodal, 15 O Outlier, 1 intuitive definition, 16 P Parameter, 2 Pareto chart, 11 Percentiles, 22 pie chart, 10 Plot box, 1 Population, 2 1

25 Q qualtitative, 5 quantitative, 5 Quartiles, 22 R Range, 18 relative frequency, 8 relative frequency distribution, 8 relative frequency histogram, 9 S Sample, 2 biased, 6 census, 6 simple random, 6 Sampling error, 6 Scatter plot, 13 simple random sample, 6 spread, 18 Standard deviation of a data set, 19 Statistic, 2 Statistics definition, 2 descriptive, 2 inferential, 2 T TI 1-Var Stats, 20 time series chart, 12 U unbiased estimator, 14 ungrouped data, 7 univariate, 3 V variability, 18 Variable, 2 Variance, 18 population, 19 sample, 19 W weighted mean, 21 Z z-score, 1 2