( ) ( ) Central Tendency. Central Tendency
|
|
- Theresa Wells
- 7 years ago
- Views:
Transcription
1 1 Central Tendency CENTRAL TENDENCY: A statistical measure that identifies a single score that is most typical or representative of the entire group Usually, a value that reflects the middle of the distribution is used, because this is where most of the scores pile up No single measure of central tendency works best in all circumstances, so there are 3 different measures -- mean, median, and mode. Each works best in a specific situation Examples: The average height of men in Montgomery County is 5 6. Central Tendency $450,000 $150,000 $100,000 The average salary in ACME Inc is $57,000. $57,000 ARITHMETICAL AVERAGE $50,000 $30,000 $37,000 MEDIAN The one in the middle ( ) 1 above him, 1 below MODE $0,000 Occurs most ( ) frequently
2 Central Tendency (Mode) MODE: The score or category that has the greatest frequency; the most common score To find the mode, simply locate the score that appears most often In a frequency distribution table, it will be the score with the largest frequency value In a frequency graph, it will be the tallest bar or point Example: A sample of class ages is given... Ages f * The age with the highest frequency is 19, with a frequency of 3; therefore, the mode is 19. Central Tendency (Mode) A distribution may have more than one mode, or peak: A distribution with modes is said to be bimodal; A distribution with more than modes is said to be multimodal Example: A sample of class ages... Age f * age and age 19 both 3 1 have a frequency of 3; if 3 this distribution were 1 1 graphed, there would be 0 1 peaks; therefore this 19 3 distribution is bimodal both and 19 are modes
3 3 Central Tendency (Mode) Advantages: Easiest to determine The only measure of central tendency that can be used with nominal (categorical) data Disadvantages: Sometimes is not a unique point in the distribution (bimodal or multimodal) Not sensitive to the location of scores in a distribution Not often used beyond the descriptive level Central Tendency (Median) MEDIAN: The score that divides the distribution exactly in half; 50% of the individuals in a distribution have scores at or below the median
4 4 Central Tendency (Median) Method when N is an odd number: List the scores from lowest to highest; the middle score on the list is the median Example: The ages of a sample of class members are 4, 18, 19,, and 0. What is the median value? List the scores from lowest to highest: 18, 19, 0,, 4 The middle score is 0 - therefore, that is the median Method when N is an even number: List scores in order from lowest to highest and locate the point halfway between the middle two scores Example: The ages of a sample of class members are 18, 19, 0,, 4 and 30. What is the median age? The scores are already listed from lowest to highest; select the middle two scores (0, ) and find the middle point: 0 + median = = 1 Central Tendency (Median) Advantages: Is less affected by extreme scores than the mean is; is better for skewed distributions Example: compare samples of class ages 1) 18, 19, 0,, 4 ) 18, 19, 0,, 47 The median in both cases is 0 - it s not thrown off by the extreme score of 47 Can be used with ordinal data The only index that can be used with open-ended distributions (distributions without a lower or upper limit for one of the categories Disadvantages: Can t be used with nominal level data Not sensitive to the location of all scores within a distribution
5 5 Central Tendency (Mean) MEAN (µ, x): The arithmetical average of the scores The amount that each individual would receive if the total (Σx) were divided up equally between everyone in the distribution Computed by adding all of the scores in the distribution and dividing that sum by the total number of scores Population mean: Sample mean: x = µ = x N x n Central Tendency (Mean) Note that, while the computations would yield the same answer, the symbols differ for a population (µ, N) and a sample (x,n) Example: x = x n = = = 0
6 6 Central Tendency (Mean) Advantages Sensitive to the location of every score in a distribution Least sensitive to sample fluctuation (if we were to take several samples, these sample means would differ less than if we compared the medians or the modes from the samples) Disadvantages May only be used with interval or ratio level data Sensitive to extreme scores, and therefore may not be desirable when working with highly skewed distributions Example: compare samples of class ages 1) 18, 19, 0,, 4 x = ) 18, 19, 0,, 47 Vs. x = 5. Central Tendency Select the method of central tendency that gives you the most information, yet is appropriate for the type of data you have. In general, use the mean if it s appropriate. If your data are skewed, or if they are not measured on an interval or ratio scale, use the median. If your data are measured on a nominal scale, the mode is the only appropriate measure of central tendency
7 7 Selecting the best measure of central tendency mean median mode mode mode mean median the mean, median, and mode for a normal distribution are the same this distribution is symmetrical, but the mean and the median are the same and it is bi-modal, it has two modes Selecting the best measure of central tendency mean median it is also possible to have a distribution with the same mean and median, but no mode In the case of a rectangular distribution
8 8 Selecting the best measure of central tendency mode mean median mean mode median for positively skewed distributions the mode would be the lowest, followed by the median then the mean for negatively skewed distributions the mean would be the lowest, followed by the median then the mode Measures of Variability VARIABILITY (a.k.a. Spread ): The degree to which scores in a distribution are spread out or clustered together This is important because we need to not only know what the average score is in a distribution, we also need to know how near or far the majority of the scores are in relation to this central value Measures of variability include: range, sums of squares (including deviation and mean deviation scores), variance, and standard deviation
9 9 Measures of Variability (Range) RANGE: The distance between the largest score and the smallest score in the distribution There are methods for computing the range: 1) Subtract the lower real limit (LRL) for the lowest score in the distribution (x min ) from the upper real limit (URL) for the highest score (x max ) Range=urlx max - lrlx min ) Subtract the minimum score from the maximum score and add 1 to the difference Range=x max - x min +1 Example: find the range for the following set of scores: 6, 6, 8, 9, 10 X max = 10 X min = 6 urlx max = 10.5 lrlx min = 5.5 Range = = 5 or Range = = 5 Measures of Variability (Range) Advantages: Easy to obtain Gives a quick approximation of variability Disadvantages: Only sensitive to the extreme scores -- insensitive to all intermediate scores For sets of scores, 1) 1, 8, 9, 9, 10, and ) 1, 3, 5, 7, 10, the range is identical although the scores are distributed very differently Substantial sample fluctuation -- can easily change from sample to sample Little used beyond the descriptive level
10 10 Measures of Variability Deviation:score - mean (X i X) Mean deviation: the average absolute deviation score X i X n Sums of Squares: The sum of the squared deviations around the mean ss = ( x µ ) Measures of Var. (Sums of Squares) Example:Two sets of quiz scores: x x- µ (x-µ) x x- µ (x-µ) X=30 0 SS=38 x=30 0 SS= µ=5.00 µ=5.00 Both distributions have the same means but the actual scores are dispersed differently
11 11 Measures of Variability (Variance) VARIANCE: The average squared deviation from the mean Provides a control for sample size (as N increases, σ SS will naturally increase) denotes population variance Formulae: SS σ = N or σ = From the previous example, 38 σ = = & 6 ( x µ ) N σ = 5 = 0.4 Measures of Var. (Standard Deviation) STANDARD DEVIATION: Measure of variability that approximates the average deviation (distance from the mean) for a given set of scores denotes population standard deviation σ Definitional formula: σ = σ For the first problem in the previous example, σ = 6.33 =.517 Our scores differ from the mean an average of.517 points
12 1 Measures of Var. (Standard Deviation) Properties of the standard deviation: Standard deviation provides a measure of the average distance from the mean When the standard deviation is small, the scores are close to the mean (the curve is narrow), and when the standard deviation is large, scores are typically spread out farther from the mean (the curve is wide) Standard deviation is a very important component of inferential statistics Measures of Var. (Standard Deviation) If a constant is added or subtracted to each score, the standard deviation does not change For example, if an instructor chooses to curve a set of test scores by adding 10 points to each score, the distance between individual scores doesn t change. All of the scores are just shifted up 10 points. The mean would increase by 10 points. However, the standard deviation remains the same
13 13 Measures of Var. (Standard Deviation) If each score in a distribution is multiplied or divided by a constant, the standard deviation of that distribution would also be multiplied or divided by the same constant Thus, if an instructor changes a 50-point exam into a 100-point exam by multiplying everyone s score by, the spread of the scores also is multiplied by. For example, on the old scale, the scores could range from 7-50 (a difference of 3 points), while on the new scale the scores range from (a difference of 46 points)(this is not the standard deviation) In this example, the standard deviation would be multiplied by. Measures of Var. (Standard Deviation) Advantages: Sensitive to the location of all scores Less sample fluctuation -- changes less from sample to sample Widely used in both descriptive and advanced statistical procedures Disadvantages: Sensitive to extreme scores -- highly skewed distributions can have a negative impact Both the range and standard deviation can only be applied to interval or ratio level scales of measurement
14 14 Measures of Variability Population vs. Sample variability The variance and standard deviation formulas we have examined so far are population formulas. These tend to underestimate the population variability when used on a sample; in other words, these are biased statistics Thus, when we are computing variances and standard deviations on samples, we correct for this bias by altering the formula; the corrected formula provides a more accurate estimate of the population values Measures of Variability Variance formula for a sample estimating a population: s SS = n 1 Standard deviation formula for a sample estimating a population: s SS = n 1 or s = s
15 15 Degrees of Freedom we know that in order to calculate variance we must know the mean ( X ) this limits the number of scores that are free to vary degrees of freedom ( df ) are defined as the number of scores in a sample that are free to vary df = n 1 where n is the number of scores in the sample Degrees of Freedom Cont. Picture Example There are five balloons: one blue, one red, one yellow, one pink, & one green. If 5 students (n=5) are each to select one balloon only 4 will have a choice of color (df=4). The last person will get whatever color is left.
16 16 Degrees of Freedom Cont. Statistical Example Given that there are 5 students ( a mean score of 10 ( X =10 ) There are four degrees of freedom df = n 1 = 5 1 = 4 n = 5 ) with In other words, four of the scores are free to vary, but the fifth is determined by the mean If we make the first four scores 9, 10, 11, & 1, then the fifth score must be 8. Statistical term Mean Variance Measures of Variability Standard deviation population value sample value x µ = N SS σ = N x = x n σ = σ s s = SS = n 1 s
17 17 Measures of Variability Choosing a Measure of Variability: When selecting a measure of variability, choose the one that gives you the most information, but is appropriate for your data situation. In general, use the standard deviation. If your data are skewed, or if they are not measured on an interval or ratio scale, use the range (Sums of squares and variance are derivatives of the standard deviation - they re used to compute the standard deviation, but provide little useful information on their own.) Standard Scores and Distributions Standard scores: Transform individual scores (raw scores) into standard (transformed) scores that give a precise description of where the scores fall within a distribution Use standard deviation units to describe the location of a score within a distribution when tests are said to be curved, the scores are transformed one way of transforming scores is to add (or subtract) a constant to each score when a constant is added or subtracted the mean will also change the same amount as the rest of the scores, but the standard deviation will be unaffected
18 18 Transformations of scores X X the mean of the X distribution is 4.5 and the standard deviation is 1.9 the mean of the X+3 distribution is 7.5 and the standard deviation is still 1.9 When a constant is added or subtracted to every score in a distribution, the shape of a distribution does not change, it simply shifts along the x-axis. Transformations of scores a second way of transforming scores is to multiply (or divide) a constant to every score in the distribution this will change the mean as well as the standard deviation the same as the rest of the scores X X(3) the mean of the X s is 4.5 and the standard deviation is 1.9 the mean of the X s multiplied by 3 is 13.5 and the standard deviation is When a constant is 3 multiplied or divided to every score in a 1 distribution, the shape 0 of a distribution changes.
19 19 Standard Scores and Distributions STANDARDIZED DISTRIBUTIONS: are composed of transformed scores with predetermined values for µ and σ (regardless of the values in the raw score distribution Examples: IQ scores are standardized with a µ=100 and σ = 15 SAT scores are standardized with a µ=500 and a σ =100 Standard Scores and Distributions Z-scores: Standard scores that specify the precise location of each raw score in a normal distribution in terms of standard deviation units Consist of parts: The sign (+ or -) indicates whether the score is located above or below the mean The magnitude of the actual number indicates how far the score is from the mean in terms of standard deviations
20 0 Examples: Standard Scores and Distributions In a distribution of test scores with µ=100, σ =15, what is the z-score for a score of 130? For a score of 85? With a score of 130, z=; it is a positive z-score because 130 is above the mean -- it s higher than 100; the magnitude is because we can add exactly standard deviations to the mean ( ) and obtain our score (130) With a score of 85, z=-1; it is below the mean (making it a negative z-score) and it is exactly 1 standard deviation from the mean (100-15=85) Standard Scores and Distributions Formula: z = x µ σ The numerator is a deviation (distance) score that indicates how far away from the mean your score of interest is, thus providing the sign (+ or -) of the z-score Dividing by σ expresses the distance score in standard deviation units (a z-score) -- this works the same way as if you knew the number of gallons a bucket held, but you wanted to express that amount in terms of quarts -- you just divide the number of gallons by four to get the number of quarts
21 1 Standard Scores and Distributions Examples: A distribution of exam scores has µ=5 and σ =3.6. You scored 9. What is your z-score? z = x µ 9 5 z = = σ 3.6 Thus, you scored 1.11 standard deviations above the mean What is your z-score if you scored on the exam? 5 z = x µ z = =. 83 σ 3.6 This time, you fell.83 standard deviations below the mean Standard Scores and Distributions You can also calculate a person s raw score (x) when you are provided with a z-score Formula: x = µ + zσ Example: A person has a z-score of 1.5 for the SAT math test (µ=500, σ =100). What is his raw score? X = (100) = = 650 Thus, his z-score indicates that his SAT math score was 650
22 Standard Scores and Distributions Characteristics of a z-score distribution: Shape: The distribution will be exactly the same shape as the distribution of raw scores Mean: The mean is always 0, regardless of the raw score distribution Standard deviation: The standard deviation of a z-score distribution will always be 1, regardless of the raw score distribution Standard Scores and Distributions Using z-scores for making comparisons: One benefit to using z-scores is that they allow comparisons between distributions with different characteristics by providing a standard metric or scale (standard deviation units) EXAMPLE: A student score 9 on a statistics exam (µ=4, σ =3), and a 50 on a biology exam (µ=50, σ =5). On which exam did the student perform better? By standardizing scores using standard deviation units, we can compare scores in completely different distributions (compare apples and oranges); Simply convert both raw scores into z-scores, then compare the z-scores to each other Statistics Biology z=(9-4)/3=1.67 z=(50-50)/5=0
23 3 Standard Scores and Distributions Other standardized distributions: Many people don t like the fact that z-score distributions have negative scores and decimal places, so they use other, similar standardized distributions which avoid the negative connotations of a - sign For example, IQ scores are viewed in terms of a standardized distribution with µ=100, s=50 t-scores distribution (which we ll learn more about later) with µ=50, s =10 Standard Scores and Distributions How do we standardize raw scores into a distribution we want? EXAMPLE: A set of exam scores have µ=43, σ =4. We would like to create a new standard distribution with µ=60, s =0. What would the new standardized value for a score of 41 on the exam? First, change the raw score into a z-score (using the procedure we just learned about) z z = = Second, change the z-score into the new standardized score x = µ + zσ Std new = µ new + zσ new Std new = x µ σ = (0) = 50
24 4 The Normal Distribution the normal distribution is not a single distribution, rather it is an infinite set of distributions that can be described using the mean ( X ) and standard deviation ( ). s the shape of the distribution describes many existing variables variables, i.e. weight The Normal Distribution by definition the area under a normal distribution = 1.0 the normal shape can also be used to determine the proportion of an area in the distribution 34% for example, the area between the mean and one standard deviation is about 34% s 1s µ 1s s
25 5 The Normal Distribution 50% 68% 13.5%.5% 34% 34% 13.5%.5% 95% each line represent 1 standard deviation, the percentages refer to the entire shaded area The Normal Distribution the areas have been calculated for all z- scores remember that z-scores transform raw scores into the number of standard deviations it is away from the mean using z-scores allows us to determine proportions or probabilities for normal distributions
26 6 The Unit Normal Table the unit normal table is a table that contains proportions in a normal distribution associated with z-score values See Table A from Pagano there are two important things to note the table includes the area in the body and in the tail there are no negative z-values The Unit Normal Table remember that the normal distribution is symmetrical, because of this the proportion will be the same whether the score is positive or negative whether the z-score is.5 or -.5, the area beyond, or the area in the tail, is still.006
27 7 The Unit Normal Table when you are dealing with the unit normal table it is sometimes confusing whether you are looking at the tail or the body, especially when you have a negative z-score perhaps the best way do deal with this confusion it to draw a picture and shade the area you are looking for The Unit Normal Table What proportion of people had a score higher than z=1.5? 1. draw a picture of the area you are looking for. draw the line where the z score is located (z=1.5) 3. shadow the area you are asked for (people who scored higher so shadow right of line) 4. then look up in tables the area in the tail for z=1.5 ANSWER: the proportion of people with a z-score higher 4 than 1.5 is.0668 Z=1.5 3 Proportion=
28 8 The Unit Normal Table You can also use the table to determine areas between scores, i.e. How many people are between z=-.5 & z=.5? 1. Draw the picture of the area you are looking for (We know that the whole area under the curve equals 1). Draw the lines where the z-scores are located 3. Shadow the area you are asked for 4. The light areas are the tails provided in the tables Area A=.31, Area B=.31. So the shadowed area equals 1- Area A - Area B = =.38 ANSWER: the proportion of people between z=-.5 & z=.5 is Z= -.5 Z=.5 Proportion=.38 3 Area B=.31 Area A= The Unit Normal Table Another method would be 1. Draw the picture of the area you are looking for. Draw the lines where the z-scores are located 3. Shadow the area for the body of z=.5 (Area C=.69) 4. Shadow the area for the tail of z= -.5 (Area D=.31) 5. Substract the tail of z=-.5 from the body of z=.5 Area C - Area D = =.38 ANSWER: the proportion of people between z=-.5 & z=.5 is Area D= Z= -.5 Z=.5 Area C=.69 Proportion=.38
DESCRIPTIVE STATISTICS. The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses.
DESCRIPTIVE STATISTICS The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses. DESCRIPTIVE VS. INFERENTIAL STATISTICS Descriptive To organize,
More informationDescriptive Statistics and Measurement Scales
Descriptive Statistics 1 Descriptive Statistics and Measurement Scales Descriptive statistics are used to describe the basic features of the data in a study. They provide simple summaries about the sample
More informationMEASURES OF VARIATION
NORMAL DISTRIBTIONS MEASURES OF VARIATION In statistics, it is important to measure the spread of data. A simple way to measure spread is to find the range. But statisticians want to know if the data are
More informationLesson 4 Measures of Central Tendency
Outline Measures of a distribution s shape -modality and skewness -the normal distribution Measures of central tendency -mean, median, and mode Skewness and Central Tendency Lesson 4 Measures of Central
More informationStatistics. Measurement. Scales of Measurement 7/18/2012
Statistics Measurement Measurement is defined as a set of rules for assigning numbers to represent objects, traits, attributes, or behaviors A variableis something that varies (eye color), a constant does
More informationFrequency Distributions
Descriptive Statistics Dr. Tom Pierce Department of Psychology Radford University Descriptive statistics comprise a collection of techniques for better understanding what the people in a group look like
More informationDescriptive Statistics. Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion
Descriptive Statistics Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion Statistics as a Tool for LIS Research Importance of statistics in research
More informationMidterm Review Problems
Midterm Review Problems October 19, 2013 1. Consider the following research title: Cooperation among nursery school children under two types of instruction. In this study, what is the independent variable?
More informationSTATS8: Introduction to Biostatistics. Data Exploration. Babak Shahbaba Department of Statistics, UCI
STATS8: Introduction to Biostatistics Data Exploration Babak Shahbaba Department of Statistics, UCI Introduction After clearly defining the scientific problem, selecting a set of representative members
More informationDescriptive Statistics
Y520 Robert S Michael Goal: Learn to calculate indicators and construct graphs that summarize and describe a large quantity of values. Using the textbook readings and other resources listed on the web
More information6.4 Normal Distribution
Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under
More informationDef: The standard normal distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1.
Lecture 6: Chapter 6: Normal Probability Distributions A normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve.
More informationBiostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY
Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY 1. Introduction Besides arriving at an appropriate expression of an average or consensus value for observations of a population, it is important to
More informationChapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs
Types of Variables Chapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs Quantitative (numerical)variables: take numerical values for which arithmetic operations make sense (addition/averaging)
More informationMeans, standard deviations and. and standard errors
CHAPTER 4 Means, standard deviations and standard errors 4.1 Introduction Change of units 4.2 Mean, median and mode Coefficient of variation 4.3 Measures of variation 4.4 Calculating the mean and standard
More information1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number
1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x - x) B. x 3 x C. 3x - x D. x - 3x 2) Write the following as an algebraic expression
More informationExercise 1.12 (Pg. 22-23)
Individuals: The objects that are described by a set of data. They may be people, animals, things, etc. (Also referred to as Cases or Records) Variables: The characteristics recorded about each individual.
More informationSTA-201-TE. 5. Measures of relationship: correlation (5%) Correlation coefficient; Pearson r; correlation and causation; proportion of common variance
Principles of Statistics STA-201-TE This TECEP is an introduction to descriptive and inferential statistics. Topics include: measures of central tendency, variability, correlation, regression, hypothesis
More informationDescriptive Statistics
Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize
More informationIntroduction to Statistics for Psychology. Quantitative Methods for Human Sciences
Introduction to Statistics for Psychology and Quantitative Methods for Human Sciences Jonathan Marchini Course Information There is website devoted to the course at http://www.stats.ox.ac.uk/ marchini/phs.html
More informationStatistics Review PSY379
Statistics Review PSY379 Basic concepts Measurement scales Populations vs. samples Continuous vs. discrete variable Independent vs. dependent variable Descriptive vs. inferential stats Common analyses
More informationNorthumberland Knowledge
Northumberland Knowledge Know Guide How to Analyse Data - November 2012 - This page has been left blank 2 About this guide The Know Guides are a suite of documents that provide useful information about
More information4.1 Exploratory Analysis: Once the data is collected and entered, the first question is: "What do the data look like?"
Data Analysis Plan The appropriate methods of data analysis are determined by your data types and variables of interest, the actual distribution of the variables, and the number of cases. Different analyses
More informationCOMPARISON MEASURES OF CENTRAL TENDENCY & VARIABILITY EXERCISE 8/5/2013. MEASURE OF CENTRAL TENDENCY: MODE (Mo) MEASURE OF CENTRAL TENDENCY: MODE (Mo)
COMPARISON MEASURES OF CENTRAL TENDENCY & VARIABILITY Prepared by: Jess Roel Q. Pesole CENTRAL TENDENCY -what is average or typical in a distribution Commonly Measures: 1. Mode. Median 3. Mean quantified
More informationChapter 2 Statistical Foundations: Descriptive Statistics
Chapter 2 Statistical Foundations: Descriptive Statistics 20 Chapter 2 Statistical Foundations: Descriptive Statistics Presented in this chapter is a discussion of the types of data and the use of frequency
More informationThe right edge of the box is the third quartile, Q 3, which is the median of the data values above the median. Maximum Median
CONDENSED LESSON 2.1 Box Plots In this lesson you will create and interpret box plots for sets of data use the interquartile range (IQR) to identify potential outliers and graph them on a modified box
More informationLesson 7 Z-Scores and Probability
Lesson 7 Z-Scores and Probability Outline Introduction Areas Under the Normal Curve Using the Z-table Converting Z-score to area -area less than z/area greater than z/area between two z-values Converting
More information7. Normal Distributions
7. Normal Distributions A. Introduction B. History C. Areas of Normal Distributions D. Standard Normal E. Exercises Most of the statistical analyses presented in this book are based on the bell-shaped
More informationMeasures of Central Tendency and Variability: Summarizing your Data for Others
Measures of Central Tendency and Variability: Summarizing your Data for Others 1 I. Measures of Central Tendency: -Allow us to summarize an entire data set with a single value (the midpoint). 1. Mode :
More informationLecture 1: Review and Exploratory Data Analysis (EDA)
Lecture 1: Review and Exploratory Data Analysis (EDA) Sandy Eckel seckel@jhsph.edu Department of Biostatistics, The Johns Hopkins University, Baltimore USA 21 April 2008 1 / 40 Course Information I Course
More informationAP Statistics Solutions to Packet 2
AP Statistics Solutions to Packet 2 The Normal Distributions Density Curves and the Normal Distribution Standard Normal Calculations HW #9 1, 2, 4, 6-8 2.1 DENSITY CURVES (a) Sketch a density curve that
More informationMEASURES OF CENTER AND SPREAD MEASURES OF CENTER 11/20/2014. What is a measure of center? a value at the center or middle of a data set
MEASURES OF CENTER AND SPREAD Mean and Median MEASURES OF CENTER What is a measure of center? a value at the center or middle of a data set Several different ways to determine the center: Mode Median Mean
More informationCALCULATIONS & STATISTICS
CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents
More informationHISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS
Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 1 of 25 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier HISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS
More informationCHAPTER THREE COMMON DESCRIPTIVE STATISTICS COMMON DESCRIPTIVE STATISTICS / 13
COMMON DESCRIPTIVE STATISTICS / 13 CHAPTER THREE COMMON DESCRIPTIVE STATISTICS The analysis of data begins with descriptive statistics such as the mean, median, mode, range, standard deviation, variance,
More informationHow To Write A Data Analysis
Mathematics Probability and Statistics Curriculum Guide Revised 2010 This page is intentionally left blank. Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction
More information3: Summary Statistics
3: Summary Statistics Notation Let s start by introducing some notation. Consider the following small data set: 4 5 30 50 8 7 4 5 The symbol n represents the sample size (n = 0). The capital letter X denotes
More informationDescriptive statistics Statistical inference statistical inference, statistical induction and inferential statistics
Descriptive statistics is the discipline of quantitatively describing the main features of a collection of data. Descriptive statistics are distinguished from inferential statistics (or inductive statistics),
More informationSection 1.3 Exercises (Solutions)
Section 1.3 Exercises (s) 1.109, 1.110, 1.111, 1.114*, 1.115, 1.119*, 1.122, 1.125, 1.127*, 1.128*, 1.131*, 1.133*, 1.135*, 1.137*, 1.139*, 1.145*, 1.146-148. 1.109 Sketch some normal curves. (a) Sketch
More informationconsider the number of math classes taken by math 150 students. how can we represent the results in one number?
ch 3: numerically summarizing data - center, spread, shape 3.1 measure of central tendency or, give me one number that represents all the data consider the number of math classes taken by math 150 students.
More informationII. DISTRIBUTIONS distribution normal distribution. standard scores
Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,
More informationEXAM #1 (Example) Instructor: Ela Jackiewicz. Relax and good luck!
STP 231 EXAM #1 (Example) Instructor: Ela Jackiewicz Honor Statement: I have neither given nor received information regarding this exam, and I will not do so until all exams have been graded and returned.
More informationEngineering Problem Solving and Excel. EGN 1006 Introduction to Engineering
Engineering Problem Solving and Excel EGN 1006 Introduction to Engineering Mathematical Solution Procedures Commonly Used in Engineering Analysis Data Analysis Techniques (Statistics) Curve Fitting techniques
More informationWeek 3&4: Z tables and the Sampling Distribution of X
Week 3&4: Z tables and the Sampling Distribution of X 2 / 36 The Standard Normal Distribution, or Z Distribution, is the distribution of a random variable, Z N(0, 1 2 ). The distribution of any other normal
More informationThe Normal Distribution
Chapter 6 The Normal Distribution 6.1 The Normal Distribution 1 6.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Recognize the normal probability distribution
More informationDescriptive statistics parameters: Measures of centrality
Descriptive statistics parameters: Measures of centrality Contents Definitions... 3 Classification of descriptive statistics parameters... 4 More about central tendency estimators... 5 Relationship between
More informationModule 3: Correlation and Covariance
Using Statistical Data to Make Decisions Module 3: Correlation and Covariance Tom Ilvento Dr. Mugdim Pašiƒ University of Delaware Sarajevo Graduate School of Business O ften our interest in data analysis
More informationMeasurement & Data Analysis. On the importance of math & measurement. Steps Involved in Doing Scientific Research. Measurement
Measurement & Data Analysis Overview of Measurement. Variability & Measurement Error.. Descriptive vs. Inferential Statistics. Descriptive Statistics. Distributions. Standardized Scores. Graphing Data.
More informationIntroduction to Environmental Statistics. The Big Picture. Populations and Samples. Sample Data. Examples of sample data
A Few Sources for Data Examples Used Introduction to Environmental Statistics Professor Jessica Utts University of California, Irvine jutts@uci.edu 1. Statistical Methods in Water Resources by D.R. Helsel
More informationDATA INTERPRETATION AND STATISTICS
PholC60 September 001 DATA INTERPRETATION AND STATISTICS Books A easy and systematic introductory text is Essentials of Medical Statistics by Betty Kirkwood, published by Blackwell at about 14. DESCRIPTIVE
More informationIntroduction; Descriptive & Univariate Statistics
Introduction; Descriptive & Univariate Statistics I. KEY COCEPTS A. Population. Definitions:. The entire set of members in a group. EXAMPLES: All U.S. citizens; all otre Dame Students. 2. All values of
More information8. THE NORMAL DISTRIBUTION
8. THE NORMAL DISTRIBUTION The normal distribution with mean μ and variance σ 2 has the following density function: The normal distribution is sometimes called a Gaussian Distribution, after its inventor,
More informationStudy Guide for the Final Exam
Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make
More informationCA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction
CA200 Quantitative Analysis for Business Decisions File name: CA200_Section_04A_StatisticsIntroduction Table of Contents 4. Introduction to Statistics... 1 4.1 Overview... 3 4.2 Discrete or continuous
More informationDensity Curve. A density curve is the graph of a continuous probability distribution. It must satisfy the following properties:
Density Curve A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: 1. The total area under the curve must equal 1. 2. Every point on the curve
More information4. Continuous Random Variables, the Pareto and Normal Distributions
4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random
More informationBNG 202 Biomechanics Lab. Descriptive statistics and probability distributions I
BNG 202 Biomechanics Lab Descriptive statistics and probability distributions I Overview The overall goal of this short course in statistics is to provide an introduction to descriptive and inferential
More informationChapter 3. The Normal Distribution
Chapter 3. The Normal Distribution Topics covered in this chapter: Z-scores Normal Probabilities Normal Percentiles Z-scores Example 3.6: The standard normal table The Problem: What proportion of observations
More informationInterpreting Data in Normal Distributions
Interpreting Data in Normal Distributions This curve is kind of a big deal. It shows the distribution of a set of test scores, the results of rolling a die a million times, the heights of people on Earth,
More informationActivities/ Resources for Unit V: Proportions, Ratios, Probability, Mean and Median
Activities/ Resources for Unit V: Proportions, Ratios, Probability, Mean and Median 58 What is a Ratio? A ratio is a comparison of two numbers. We generally separate the two numbers in the ratio with a
More informationSession 7 Bivariate Data and Analysis
Session 7 Bivariate Data and Analysis Key Terms for This Session Previously Introduced mean standard deviation New in This Session association bivariate analysis contingency table co-variation least squares
More informationDescribing Data: Measures of Central Tendency and Dispersion
100 Part 2 / Basic Tools of Research: Sampling, Measurement, Distributions, and Descriptive Statistics Chapter 8 Describing Data: Measures of Central Tendency and Dispersion In the previous chapter we
More informationSummarizing and Displaying Categorical Data
Summarizing and Displaying Categorical Data Categorical data can be summarized in a frequency distribution which counts the number of cases, or frequency, that fall into each category, or a relative frequency
More informationMBA 611 STATISTICS AND QUANTITATIVE METHODS
MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 1-11) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain
More informationThe Big Picture. Describing Data: Categorical and Quantitative Variables Population. Descriptive Statistics. Community Coalitions (n = 175)
Describing Data: Categorical and Quantitative Variables Population The Big Picture Sampling Statistical Inference Sample Exploratory Data Analysis Descriptive Statistics In order to make sense of data,
More informationLecture 2: Descriptive Statistics and Exploratory Data Analysis
Lecture 2: Descriptive Statistics and Exploratory Data Analysis Further Thoughts on Experimental Design 16 Individuals (8 each from two populations) with replicates Pop 1 Pop 2 Randomly sample 4 individuals
More informationCenter: Finding the Median. Median. Spread: Home on the Range. Center: Finding the Median (cont.)
Center: Finding the Median When we think of a typical value, we usually look for the center of the distribution. For a unimodal, symmetric distribution, it s easy to find the center it s just the center
More informationData Exploration Data Visualization
Data Exploration Data Visualization What is data exploration? A preliminary exploration of the data to better understand its characteristics. Key motivations of data exploration include Helping to select
More informationIntroduction to Quantitative Methods
Introduction to Quantitative Methods October 15, 2009 Contents 1 Definition of Key Terms 2 2 Descriptive Statistics 3 2.1 Frequency Tables......................... 4 2.2 Measures of Central Tendencies.................
More informationTest 4 Sample Problem Solutions, 27.58 = 27 47 100, 7 5, 1 6. 5 = 14 10 = 1.4. Moving the decimal two spots to the left gives
Test 4 Sample Problem Solutions Convert from a decimal to a fraction: 0.023, 27.58, 0.777... For the first two we have 0.023 = 23 58, 27.58 = 27 1000 100. For the last, if we set x = 0.777..., then 10x
More informationDESCRIPTIVE STATISTICS & DATA PRESENTATION*
Level 1 Level 2 Level 3 Level 4 0 0 0 0 evel 1 evel 2 evel 3 Level 4 DESCRIPTIVE STATISTICS & DATA PRESENTATION* Created for Psychology 41, Research Methods by Barbara Sommer, PhD Psychology Department
More information3. What is the difference between variance and standard deviation? 5. If I add 2 to all my observations, how variance and mean will vary?
Variance, Standard deviation Exercises: 1. What does variance measure? 2. How do we compute a variance? 3. What is the difference between variance and standard deviation? 4. What is the meaning of the
More informationChapter 2: Descriptive Statistics
Chapter 2: Descriptive Statistics **This chapter corresponds to chapters 2 ( Means to an End ) and 3 ( Vive la Difference ) of your book. What it is: Descriptive statistics are values that describe the
More informationWISE Power Tutorial All Exercises
ame Date Class WISE Power Tutorial All Exercises Power: The B.E.A.. Mnemonic Four interrelated features of power can be summarized using BEA B Beta Error (Power = 1 Beta Error): Beta error (or Type II
More informationModule 4: Data Exploration
Module 4: Data Exploration Now that you have your data downloaded from the Streams Project database, the detective work can begin! Before computing any advanced statistics, we will first use descriptive
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationLecture Notes Module 1
Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific
More informationThe Dummy s Guide to Data Analysis Using SPSS
The Dummy s Guide to Data Analysis Using SPSS Mathematics 57 Scripps College Amy Gamble April, 2001 Amy Gamble 4/30/01 All Rights Rerserved TABLE OF CONTENTS PAGE Helpful Hints for All Tests...1 Tests
More informationCalculation example mean, median, midrange, mode, variance, and standard deviation for raw and grouped data
Calculation example mean, median, midrange, mode, variance, and standard deviation for raw and grouped data Raw data: 7, 8, 6, 3, 5, 5, 1, 6, 4, 10 Sorted data: 1, 3, 4, 5, 5, 6, 6, 7, 8, 10 Number of
More informationStatistics I for QBIC. Contents and Objectives. Chapters 1 7. Revised: August 2013
Statistics I for QBIC Text Book: Biostatistics, 10 th edition, by Daniel & Cross Contents and Objectives Chapters 1 7 Revised: August 2013 Chapter 1: Nature of Statistics (sections 1.1-1.6) Objectives
More informationDATA COLLECTION AND ANALYSIS
DATA COLLECTION AND ANALYSIS Quality Education for Minorities (QEM) Network HBCU-UP Fundamentals of Education Research Workshop Gerunda B. Hughes, Ph.D. August 23, 2013 Objectives of the Discussion 2 Discuss
More information2. Filling Data Gaps, Data validation & Descriptive Statistics
2. Filling Data Gaps, Data validation & Descriptive Statistics Dr. Prasad Modak Background Data collected from field may suffer from these problems Data may contain gaps ( = no readings during this period)
More informationWhy Taking This Course? Course Introduction, Descriptive Statistics and Data Visualization. Learning Goals. GENOME 560, Spring 2012
Why Taking This Course? Course Introduction, Descriptive Statistics and Data Visualization GENOME 560, Spring 2012 Data are interesting because they help us understand the world Genomics: Massive Amounts
More informationExploratory Data Analysis
Exploratory Data Analysis Johannes Schauer johannes.schauer@tugraz.at Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Introduction
More informationPie Charts. proportion of ice-cream flavors sold annually by a given brand. AMS-5: Statistics. Cherry. Cherry. Blueberry. Blueberry. Apple.
Graphical Representations of Data, Mean, Median and Standard Deviation In this class we will consider graphical representations of the distribution of a set of data. The goal is to identify the range of
More informationSTT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables
Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random
More information4. Descriptive Statistics: Measures of Variability and Central Tendency
4. Descriptive Statistics: Measures of Variability and Central Tendency Objectives Calculate descriptive for continuous and categorical data Edit output tables Although measures of central tendency and
More informationThe Normal Distribution
The Normal Distribution Continuous Distributions A continuous random variable is a variable whose possible values form some interval of numbers. Typically, a continuous variable involves a measurement
More informationz-scores AND THE NORMAL CURVE MODEL
z-scores AND THE NORMAL CURVE MODEL 1 Understanding z-scores 2 z-scores A z-score is a location on the distribution. A z- score also automatically communicates the raw score s distance from the mean A
More informationINTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA)
INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA) As with other parametric statistics, we begin the one-way ANOVA with a test of the underlying assumptions. Our first assumption is the assumption of
More informationPractice#1(chapter1,2) Name
Practice#1(chapter1,2) Name Solve the problem. 1) The average age of the students in a statistics class is 22 years. Does this statement describe descriptive or inferential statistics? A) inferential statistics
More informationIntroduction to Statistics and Quantitative Research Methods
Introduction to Statistics and Quantitative Research Methods Purpose of Presentation To aid in the understanding of basic statistics, including terminology, common terms, and common statistical methods.
More informationStudy Guide for Essentials of Statistics for the Social and Behavioral Sciences by Barry H. Cohen and R. Brooke Lea. Chapter 1
Distributions Study Guide for Essentials of Statistics for the Social and Behavioral Sciences by Barry H. Cohen and R. Brooke Lea Chapter 1 Guidelines for Frequency Distributions The procedure for constructing
More informationProbability Distributions
Learning Objectives Probability Distributions Section 1: How Can We Summarize Possible Outcomes and Their Probabilities? 1. Random variable 2. Probability distributions for discrete random variables 3.
More informationTHE BINOMIAL DISTRIBUTION & PROBABILITY
REVISION SHEET STATISTICS 1 (MEI) THE BINOMIAL DISTRIBUTION & PROBABILITY The main ideas in this chapter are Probabilities based on selecting or arranging objects Probabilities based on the binomial distribution
More informationNormal distribution. ) 2 /2σ. 2π σ
Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a
More informationExploratory data analysis (Chapter 2) Fall 2011
Exploratory data analysis (Chapter 2) Fall 2011 Data Examples Example 1: Survey Data 1 Data collected from a Stat 371 class in Fall 2005 2 They answered questions about their: gender, major, year in school,
More informationFinal Exam Practice Problem Answers
Final Exam Practice Problem Answers The following data set consists of data gathered from 77 popular breakfast cereals. The variables in the data set are as follows: Brand: The brand name of the cereal
More informationStatistical Confidence Calculations
Statistical Confidence Calculations Statistical Methodology Omniture Test&Target utilizes standard statistics to calculate confidence, confidence intervals, and lift for each campaign. The student s T
More informationThe GED math test gives you a page of math formulas that
Math Smart 643 The GED Math Formulas The GED math test gives you a page of math formulas that you can use on the test, but just seeing the formulas doesn t do you any good. The important thing is understanding
More information