# Measures of Central Tendency. There are different types of averages, each has its own advantages and disadvantages.

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Measures of Central Tendency According to Prof Bowley Measures of central tendency (averages) are statistical constants which enable us to comprehend in a single effort the significance of the whole. The main objectives of Measure of Central Tendency are 1) To condense data in a single value. 2) To facilitate comparisons between data. There are different types of averages, each has its own advantages and disadvantages. Requisites of a Good Measure of Central Tendency: 1. It should be rigidly defined. 2. It should be simple to understand & easy to calculate. 3. It should be based upon all values of given data. 4. It should be capable of further mathematical treatment. 5. It should have sampling stability. 6. It should be not be unduly affected by extreme values. Measure of Central Tendency Locational (positional ) average Mathematical Average Partition values Mode Arithmetic Geometric Harmonic Median Quartiles Deciles Percentiles Mean Mean Mean Partition values: The points which divide the data in to equal parts are called Partition values. Median: The point or the value which divides the data in to two equal parts., or when the data is arranged in numerical order The data must be ranked (sorted in ascending order) first. The median is the number in the middle. Depending on the data size we define median as:

2 It is the middle value when data size N is odd. It is the mean of the middle two values, when data size N is even. Ungrouped Frequency Distribution Find the cumulative frequencies for the data. The value of the variable corresponding to which a cumulative frequency is greater than (N+1)/2 for the first time. (Where N is the total number of observations.) Example 1: Find the median for the following frequency distribution. X Freq Solution: Calculate cumulative frequencies less than type. X Freq Cum freq N=120, ( N+1)/2=60.5 this value is first exceeded by cumulative frequency 65, this value is corresponding to X-value 5, hence median is 5 Grouped Frequency Distribution First obtain the cumulative frequencies for the data. Then mark the class corresponding to which a cumulative frequency is greater than (N)/2 for the first time. (N is the total number of observations.) Then that class is median class. Then median is evaluated by interpolation formula. Where l 1 = lower limit of the median class, l 2 = upper limit of the median class N= Number of observations. cf = cumulative frequency of the class proceeding to the median class. f m = frequency of the median class. Quartiles : The data can be divided in to four equal parts by three points. These three points are known as quartiles. The quartiles are denoted by Q i, i = 1,2,3

3 Qi is the value corresponding to (in/4) th order. observation after arranging the data in the increasing For grouped data : First obtain the cumulative frequencies for the data. Then mark the class corresponding to which a cumulative frequency is greater than (in)/4 for the first time. (Where N is total number of observations.). Then that class is Qi class. Then Qi is evaluated by interpolation formula. i= 1, 2, 3 Where l 1 = lower limit of the Qi class, l 2 = upper limit of the Qi class N= Number of observations. cf = cumulative frequency of the class proceeding to the Qi class. f q = frequency of the Qi class. Deciles are nine points which divided the data in to ten equal parts. Di is the value corresponding to (in/10) th order. observation after arranging the data in the increasing For grouped data :First obtain the cumulative frequencies for the data. Then mark the class corresponding to which a cumulative frequency is greater than (in)/10 for the first time. (Where N is total number of observations.). Then that class is Di class. Then Di is evaluated by interpolation formula. i= 1, 2, 10. Where l 1 = lower limit of the Di class, l 2 = upper limit of the Di class N= Number of observations. cf = cumulative frequency of the class proceeding to the Di class. f d = frequency of the Di class. Percentiles are ninety-nine points which divided the data in to hundred equal parts. Pi is the value corresponding to (in/100) th order. observation after arranging the data in the increasing For grouped data : First obtain the cumulative frequencies for the data. Then mark the class corresponding to which a cumulative frequency is greater than (in)/100 for the first time. (Where N

4 is total number of observations.) Then that class is Pi class. Then Pi is evaluated by interpolation formula. Where l 1 = lower limit of the Pi class, l 2 = upper limit of the Pi class N= Number of observations. cf = cumulative frequency of the class proceeding to the Pi class. f p = frequency of the Pi class. Graphical method for locating partition values: These partition values can be located graphically by using ogives. The point of intersection of both ogives is median. To locate quartiles, mark N/4 on Y- axis, from that point draw a line parallel to X-axis, it cuts less than type ogive at Q 1 and intersects greater than or equal to curve at Q 3. To locate Di mark in/10 on Y-axis, from that point draw line parallel to X-axis, it intersects less than type curve at Di. Similarly to locate Pi mark in/100 on Y-axis, from that point draw line parallel to X-axis, it intersects less than type curve at Pi. Example 2. Find the median Daily wages in Rs. No of workers Solution : To locate median class we have to calculate cumulative frequencies. Daily wages in Rs No of workers Cum Freq N=50, N/2= 25 so median class is

5 Example 3 : Find the median, Q 1, D8,P65 from the following data. Marks No of Students Solution : To locate median class we have to calculate cumulative frequencies. Marks No of Students Cumulative freq Here N=50 so N/2=25, hence median class is Here N=50 so N/4=12.5, hence Q1class is Here N=50 so 8*N/10=40, hence D8 class is Here N=50 so 65*N/100=32.5, hence P65 class is Use the median to describe the middle of a set of data that does have an outlier.

6 Merits of Median 1. It is rigidly defined. 2. It is easy to understand & easy to calculate. 3. It is not affected by extreme values. 4. Even if extreme values are not known median can be calculated. 5. It can be located just by inspection in many cases. 6. It can be located graphically. 7. It is not much affected by sampling fluctuations. 8. It can be calculated for data based on ordinal scale. Demerits of Median 1. It is not based upon all values of the given data. 2. For larger data size the arrangement of data in the increasing order is difficult process. 3. It is not capable of further mathematical treatment. 4. It is insensitive to some changes in the data values. MODE The mode is the most frequent data value. Mode is the value of the variable which is predominant in the given data series. Thus in case of discrete frequency distribution, mode is the value corresponding to maximum frequency. Sometimes there may be no single mode if no one value appears more than any other. There may also be two modes (bimodal), three modes (trimodal), or more than three modes (multi-modal). For grouped frequency distributions, the modal class is the class with the largest frequency. After identifying modal class mode is evaluated by using interpolated formula. This formula is applicable when classes are of equal width. Where l 1 = lower limit of the modal class, l 2 = upper limit of the modal class d1 =fm-f 0 and d2=fm-f 1 where fm= frequency of the modal class, f 0 = frequency of the class preceding to the modal class, f 1 = frequency of the class succeeding to the modal class. Mode can be located graphically by drawing histogram.

7 Steps: 1) Draw histogram 2) Locate modal class (highest bar of the histogram 3) Join diagonally the upper end points of the end points of the highest bar to the adjacent bars. 4) Mark the point of intersection of the diagonals. 5) Draw the perpendicular from this point on the X-axis. 6) The point where the perpendicular meets X-axis gives the modal value. Example 4: Find the mode Classes Frequency Modal class : d1= fm-f0= 27-18=9 d2= fm-f1= 27-20= = Mode Use the mode when the data is non-numeric or when asked to choose the most popular item. Merits of Mode 1. It is easy to understand & easy to calculate.

8 2. It is not affected by extreme values or sampling fluctuations. 3. Even if extreme values are not known mode can be calculated. 4. It can be located just by inspection in many cases. 5. It is always present within the data. 6. It can be located graphically. 7. It is applicable for both qualitative and quantitative data. Demerits of Mode 1. It is not rigidly defined. 2. It is not based upon all values of the given data. 3. It is not capable of further mathematical treatment. Arithmetic Mean This is what people usually intend when they say "average" Sample mean: If X 1, X 2, X n are data values then arithmetic mean is given by Frequency Distribution: Let X 1, X 2, X n are class marks and the corresponding frequencies are f 1, f 2, f n, then arithmetic mean is given by Example 5 : The Marks obtained in 10 class tests are 25, 20, 20, 9, 16, 10, 21, 12, 8, 13. The mean = Example 6 : Find the mean Xi Freq=fi Then N= fi = 60, and fi Xi= 731

9 Example 7 : The following data represents income distribution of 100 families, Calculate mean income of 100 families. Income in 00 Rs. No. of families Solution: We have Income in 00 Rs. Class Mark Xi No. of families fi We get N= fi =100, fi Xi= 6330 Mean = Properties of Mean: 1) Effect of shift of origin and scale. If X 1, X 2 X n are given values. New values U are obtained by shifting the origin to a and changing scale by h then Mean= 2) Algebraic sum of deviations of set of values taken from their mean is zero. a. If X 1, X 2 X n are given values then b. If X 1, X 2 X n are given values with corresponding frequencies f1, f2, fn then 3) The sum of squares of deviation of set of values about its mean is minimum. 4) If i= 1,2..n where A

10 5) If are the means of two sets of values containing n 1 and n 2 observations respectively then the mean of the combined data is given by This formula can be extended for k sets of data values as Merits of Mean 1. It is rigidly defined. 2. It is easy to understand & easy to calculate. 3. It is based upon all values of the given data. 4. It is capable of further mathematical treatment. 5. It is not much affected by sampling fluctuations. Demerits of Mean 1. It cannot be calculated if any observations are missing. 2. It cannot be calculated for the data with open end classes. 3. It is affected by extreme values. 4. It cannot be located graphically. 5. It may be number which is not present in the data. 6. It can be calculated for the data representing qualitative characteristic. Empirical formula: For symmetric distribution Mean, Median and Mode coincide. If the distribution is moderately asymmetrical the Mean, Median and Mode satisfy the following relationship Mean- Mode =3(Mean- Median) Or Mode=3Median-2Mean Weighted mean : If X 1, X 2 X n are given values with corresponding weights W 1, W 2, W n then the weighted mean is given by The mean of a frequency distribution is also the weighted mean. Use the mean to describe the middle of a set of data that does not have an outlier. Geometric Mean: a. If X 1, X 2 X n are given values then

11 Or GM= antilog b. If X 1, X 2 X n are given values with corresponding frequencies f1, f2, fn then if N= fi Merits of Geometric Mean GM= antilog 1. It is based upon all values of the given data. 2. It is capable of further mathematical treatment. 3. It is not much affected by sampling fluctuations. Demerits of Geometric Mean 1. It is not easy to understand & not easy to calculate 2. It is not well defined. 3. If anyone data value is zero then GM is zero. 4. It cannot be calculated if any observations are missing. 5. It cannot be calculated for the data with open end classes. 6. It is affected by extreme values. 7. It cannot be located graphically. 8. It may be number which is not present in the data. 9. It cannot be calculated for the data representing qualitative characteristic Harmonic Mean: a. If X 1, X 2 X n are given values then Harmonic Mean is given by b. If X 1, X 2 X n are given values with corresponding frequencies f1, f2, fn then Harmonic Mean given by if N= fi

12 Merits of Harmonic Mean 1. It is rigidly defined. 2. It is easy to understand & easy to calculate. 3. It is based upon all values of the given data. 4. It is capable of further mathematical treatment. 5. It is not much affected by sampling fluctuations. Demerits of Harmonic Mean 1. It is not easy to understand & not easy to calculate. 2. It cannot be calculated if any observations are missing. 3. It cannot be calculated for the data with open end classes. 4. It is usually not a good representative of the data. 5. It is affected by extreme values. 6. It cannot be located graphically. 7. It may be number which is not present in the data. 8. It can be calculated for the data representing qualitative characteristic. Selection of an average: No single average can be regarded as the best or most suitable under all circumstances. Each average has its merits and demerits and its own particular field of importance and utility. A proper selection of an average depends on the 1) nature of the data and 2) purpose of enquiry or requirement of the data. A.M. satisfies almost all the requisites of a good average and hence can be regarded as the best average but it cannot be used 1) in case of highly skewed data. 2) in case of uneven or irregular spread of the data. 3) in open end distributions. 4) When average growth or average speed is required. 5) When there are extreme values in the data. Except in these cases AM is widely used in practice. Median: is the best average in open end distributions or in distributions which give highly skew or j or reverse j type frequency curves. In such cases A.M. gives unnecessarily high or low value whereas median gives a more representative value. But in case of fairly symmetric distribution there is nothing to choose between mean, median and mode, as they are very close to each other. Mode : is especially useful to describe qualitative data. According to Freunel and Williams, consumer preferences for different kinds of products can be compared using modal preferences as we cannot compute mean or median. Mode can best describe the average size of shoes or shirts.

13 G.M. is useful to average relative changes, averaging ratios and percentages. It is theoretically the best average for construction of index number. But it should not be used for measuring absolute changes. H.M. is useful in problems where values of a variable are compared with a constant quantity of another variable like time, distance travelled within a given time, quantities purchased or sold over a unit. In general we can say that A.M. is the best of all averages and other averages may be used under special circumstances.

### MCQ S OF MEASURES OF CENTRAL TENDENCY

MCQ S OF MEASURES OF CENTRAL TENDENCY MCQ No 3.1 Any measure indicating the centre of a set of data, arranged in an increasing or decreasing order of magnitude, is called a measure of: (a) Skewness (b)

### Grouped Data The mean for grouped data is obtained from the following formula:

Lecture.4 Measures of averages - Mean median mode geometric mean harmonic mean computation of the above statistics for raw and grouped data - merits and demerits - measures of location percentiles quartiles

### Measures of Central Tendency

CHAPTER Measures of Central Tendency Studying this chapter should enable you to: understand the need for summarising a set of data by one single number; recognise and distinguish between the different

### Session 1.6 Measures of Central Tendency

Session 1.6 Measures of Central Tendency Measures of location (Indices of central tendency) These indices locate the center of the frequency distribution curve. The mode, median, and mean are three indices

### 1 Measures of Central Tendency

CHAPTER 1 Measures of Central Tendency 1.1 INTRODUCTION: STATISTICS It is that branch of science which deals with the collection of data, organising summarising, presenting and analysing data and making

### MEASURES OF CENTRAL TENDENCY INTRODUCTION

2 MEASURES OF CENTRAL TENDENCY INTRODUCTION A requency distribution, in general, shows clustering o the data around some central value. Finding o this central value or the average is o importance, as it

### Numerical Summarization of Data OPRE 6301

Numerical Summarization of Data OPRE 6301 Motivation... In the previous session, we used graphical techniques to describe data. For example: While this histogram provides useful insight, other interesting

### CHINHOYI UNIVERSITY OF TECHNOLOGY

CHINHOYI UNIVERSITY OF TECHNOLOGY SCHOOL OF NATURAL SCIENCES AND MATHEMATICS DEPARTMENT OF MATHEMATICS MEASURES OF CENTRAL TENDENCY AND DISPERSION INTRODUCTION From the previous unit, the Graphical displays

CHAPTER 6 STATISTICS KEY POINTS 1. The mean for grouped data can be found by : fixi (i) The direct method X. (ii) The assumed mean method X a, (iii) here d i = x i a. The step deviation method fiui i,

### 10-3 Measures of Central Tendency and Variation

10-3 Measures of Central Tendency and Variation So far, we have discussed some graphical methods of data description. Now, we will investigate how statements of central tendency and variation can be used.

### Frequency Distributions

Displaying Data Frequency Distributions After collecting data, the first task for a researcher is to organize and summarize the data to get a general overview of the results. Remember, this is the goal

### A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes

A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes together with the number of data values from the set that

### We will use the following data sets to illustrate measures of center. DATA SET 1 The following are test scores from a class of 20 students:

MODE The mode of the sample is the value of the variable having the greatest frequency. Example: Obtain the mode for Data Set 1 77 For a grouped frequency distribution, the modal class is the class having

### not to be republished NCERT Measures of Central Tendency

You have learnt in previous chapter that organising and presenting data makes them comprehensible. It facilitates data processing. A number of statistical techniques are used to analyse the data. In this

### Descriptive 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

### 2. Describing Data. We consider 1. Graphical methods 2. Numerical methods 1 / 56

2. Describing Data We consider 1. Graphical methods 2. Numerical methods 1 / 56 General Use of Graphical and Numerical Methods Graphical methods can be used to visually and qualitatively present data and

### 1) 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

### Statistical Concepts and Market Return

Statistical Concepts and Market Return 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 2 2. Some Fundamental Concepts... 2 3. Summarizing Data Using Frequency Distributions...

### LearnStat MEASURES OF CENTRAL TENDENCY, Learning Statistics the Easy Way. Session on BUREAU OF LABOR AND EMPLOYMENT STATISTICS

LearnStat t Learning Statistics the Easy Way Session on MEASURES OF CENTRAL TENDENCY, DISPERSION AND SKEWNESS MEASURES OF CENTRAL TENDENCY, DISPERSION AND SKEWNESS OBJECTIVES At the end of the session,

### Report of for Chapter 2 pretest

Report of for Chapter 2 pretest Exam: Chapter 2 pretest Category: Organizing and Graphing Data 1. "For our study of driving habits, we recorded the speed of every fifth vehicle on Drury Lane. Nearly every

### F. Farrokhyar, MPhil, PhD, PDoc

Learning objectives Descriptive Statistics F. Farrokhyar, MPhil, PhD, PDoc To recognize different types of variables To learn how to appropriately explore your data How to display data using graphs How

### Chapter 3: Data Description Numerical Methods

Chapter 3: Data Description Numerical Methods Learning Objectives Upon successful completion of Chapter 3, you will be able to: Summarize data using measures of central tendency, such as the mean, median,

### 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,

### 2.0 Lesson Plan. Answer Questions. Summary Statistics. Histograms. The Normal Distribution. Using the Standard Normal Table

2.0 Lesson Plan Answer Questions 1 Summary Statistics Histograms The Normal Distribution Using the Standard Normal Table 2. Summary Statistics Given a collection of data, one needs to find representations

### Chapter 2: Exploring Data with Graphs and Numerical Summaries. Graphical Measures- Graphs are used to describe the shape of a data set.

Page 1 of 16 Chapter 2: Exploring Data with Graphs and Numerical Summaries Graphical Measures- Graphs are used to describe the shape of a data set. Section 1: Types of Variables In general, variable can

### Univariate Descriptive Statistics

Univariate Descriptive Statistics Displays: pie charts, bar graphs, box plots, histograms, density estimates, dot plots, stemleaf plots, tables, lists. Example: sea urchin sizes Boxplot Histogram Urchin

### Central Tendency. n Measures of Central Tendency: n Mean. n Median. n Mode

Central Tendency Central Tendency n A single summary score that best describes the central location of an entire distribution of scores. n Measures of Central Tendency: n Mean n The sum of all scores divided

### Desciptive Statistics Qualitative data Quantitative data Graphical methods Numerical methods

Desciptive Statistics Qualitative data Quantitative data Graphical methods Numerical methods Qualitative data Data are classified in categories Non numerical (although may be numerically codified) Elements

### Descriptive 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

### 18.2. STATISTICS 2 (Measures of central tendency) A.J.Hobson

JUST THE MATHS SLIDES NUMBER 18.2 STATISTICS 2 (Measures of central tendency) by A.J.Hobson 18.2.1 Introduction 18.2.2 The arithmetic mean (by coding) 18.2.3 The median 18.2.4 The mode 18.2.5 Quantiles

### 4. Introduction to Statistics

Statistics for Engineers 4-1 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation

### Chapter 7 What to do when you have the data

Chapter 7 What to do when you have the data We saw in the previous chapters how to collect data. We will spend the rest of this course looking at how to analyse the data that we have collected. Stem and

### Chapter 2. Objectives. Tabulate Qualitative Data. Frequency Table. Descriptive Statistics: Organizing, Displaying and Summarizing Data.

Objectives Chapter Descriptive Statistics: Organizing, Displaying and Summarizing Data Student should be able to Organize data Tabulate data into frequency/relative frequency tables Display data graphically

### Chapter 3 Descriptive Statistics: Numerical Measures. Learning objectives

Chapter 3 Descriptive Statistics: Numerical Measures Slide 1 Learning objectives 1. Single variable Part I (Basic) 1.1. How to calculate and use the measures of location 1.. How to calculate and use the

### Descriptive Statistics. Frequency Distributions and Their Graphs 2.1. Frequency Distributions. Chapter 2

Chapter Descriptive Statistics.1 Frequency Distributions and Their Graphs Frequency Distributions A frequency distribution is a table that shows classes or intervals of data with a count of the number

### Organizing & Graphing Data

AGSC 320 Statistical Methods Organizing & Graphing Data 1 DATA Numerical representation of reality Raw data: Data recorded in the sequence in which they are collected and before any processing Qualitative

### 7 th Grade Pre Algebra A Vocabulary Chronological List

SUM sum the answer to an addition problem Ex. 4 + 5 = 9 The sum is 9. DIFFERENCE difference the answer to a subtraction problem Ex. 8 2 = 6 The difference is 6. PRODUCT product the answer to a multiplication

### GCSE HIGHER Statistics Key Facts

GCSE HIGHER Statistics Key Facts Collecting Data When writing questions for questionnaires, always ensure that: 1. the question is worded so that it will allow the recipient to give you the information

### CHAPTER 3 CENTRAL TENDENCY ANALYSES

CHAPTER 3 CENTRAL TENDENCY ANALYSES The next concept in the sequential statistical steps approach is calculating measures of central tendency. Measures of central tendency represent some of the most simple

### Unit-4 Measures of Central Tendency B.A.III(HONS)

Unit-4 Measures of Central Tendency B.A.III(HONS) INTRODUCTION Measures that reflect the average characteristics of a frequency distribution are referred to as measures of central tendency. Mean, median

### 1 Measures for location and dispersion of a sample

Statistical Geophysics WS 2008/09 7..2008 Christian Heumann und Helmut Küchenhoff Measures for location and dispersion of a sample Measures for location and dispersion of a sample In the following: Variable

### Descriptive Statistics. Understanding Data: Categorical Variables. Descriptive Statistics. Dataset: Shellfish Contamination

Descriptive Statistics Understanding Data: Dataset: Shellfish Contamination Location Year Species Species2 Method Metals Cadmium (mg kg - ) Chromium (mg kg - ) Copper (mg kg - ) Lead (mg kg - ) Mercury

### The 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

### STATISTICS FOR PSYCH MATH REVIEW GUIDE

STATISTICS FOR PSYCH MATH REVIEW GUIDE ORDER OF OPERATIONS Although remembering the order of operations as BEDMAS may seem simple, it is definitely worth reviewing in a new context such as statistics formulae.

### Introduction 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

### The Big 50 Revision Guidelines for S1

The Big 50 Revision Guidelines for S1 If you can understand all of these you ll do very well 1. Know what is meant by a statistical model and the Modelling cycle of continuous refinement 2. Understand

### Table 2-1. Sucrose concentration (% fresh wt.) of 100 sugar beet roots. Beet No. % Sucrose. Beet No.

Chapter 2. DATA EXPLORATION AND SUMMARIZATION 2.1 Frequency Distributions Commonly, people refer to a population as the number of individuals in a city or county, for example, all the people in California.

### Lesson 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

### Data 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

### TYPES OF DATA TYPES OF VARIABLES

TYPES OF DATA Univariate data Examines the distribution features of one variable. Bivariate data Explores the relationship between two variables. Univariate and bivariate analysis will be revised separately.

### CHAPTER 86 MEAN, MEDIAN, MODE AND STANDARD DEVIATION

CHAPTER 86 MEAN, MEDIAN, MODE AND STANDARD DEVIATION EXERCISE 36 Page 919 1. Determine the mean, median and modal values for the set: {3, 8, 10, 7, 5, 14,, 9, 8} Mean = 3+ 8 + 10 + 7 + 5 + 14 + + 9 + 8

### Center: 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

### Data Mining Part 2. Data Understanding and Preparation 2.1 Data Understanding Spring 2010

Data Mining Part 2. and Preparation 2.1 Spring 2010 Instructor: Dr. Masoud Yaghini Introduction Outline Introduction Measuring the Central Tendency Measuring the Dispersion of Data Graphic Displays References

### Chapter 3: Central Tendency

Chapter 3: Central Tendency Central Tendency In general terms, central tendency is a statistical measure that determines a single value that accurately describes the center of the distribution and represents

### MBA 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

### Descriptive 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

### 1.5 NUMERICAL REPRESENTATION OF DATA (Sample Statistics)

1.5 NUMERICAL REPRESENTATION OF DATA (Sample Statistics) As well as displaying data graphically we will often wish to summarise it numerically particularly if we wish to compare two or more data sets.

### Central Tendency and Variation

Contents 5 Central Tendency and Variation 161 5.1 Introduction............................ 161 5.2 The Mode............................. 163 5.2.1 Mode for Ungrouped Data................ 163 5.2.2 Mode

### x Measures of Central Tendency for Ungrouped Data Chapter 3 Numerical Descriptive Measures Example 3-1 Example 3-1: Solution

Chapter 3 umerical Descriptive Measures 3.1 Measures of Central Tendency for Ungrouped Data 3. Measures of Dispersion for Ungrouped Data 3.3 Mean, Variance, and Standard Deviation for Grouped Data 3.4

### In this course, we will consider the various possible types of presentation of data and justification for their use in given situations.

PRESENTATION OF DATA 1.1 INTRODUCTION Once data has been collected, it has to be classified and organised in such a way that it becomes easily readable and interpretable, that is, converted to information.

### Graphical and Tabular. Summarization of Data OPRE 6301

Graphical and Tabular Summarization of Data OPRE 6301 Introduction and Re-cap... Descriptive statistics involves arranging, summarizing, and presenting a set of data in such a way that useful information

### 13.2 Measures of Central Tendency

13.2 Measures of Central Tendency Measures of Central Tendency For a given set of numbers, it may be desirable to have a single number to serve as a kind of representative value around which all the numbers

### Lesson outline 1: Mean, median, mode

4. Give 5 different numbers such that their average is 21. The numbers are: I found these numbers by: 5. A median of a set of scores is the value in the middle when the scores are placed in order. In case

### Quantitative Data Analysis: Choosing a statistical test Prepared by the Office of Planning, Assessment, Research and Quality

Quantitative Data Analysis: Choosing a statistical test Prepared by the Office of Planning, Assessment, Research and Quality 1 To help choose which type of quantitative data analysis to use either before

### 32 Measures of Central Tendency and Dispersion

32 Measures of Central Tendency and Dispersion In this section we discuss two important aspects of data which are its center and its spread. The mean, median, and the mode are measures of central tendency

### Probability and Statistics Vocabulary List (Definitions for Middle School Teachers)

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence

### 3: 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

### Research Variables. Measurement. Scales of Measurement. Chapter 4: Data & the Nature of Measurement

Chapter 4: Data & the Nature of Graziano, Raulin. Research Methods, a Process of Inquiry Presented by Dustin Adams Research Variables Variable Any characteristic that can take more than one form or value.

### Describing Data. We find the position of the central observation using the formula: position number =

HOSP 1207 (Business Stats) Learning Centre Describing Data This worksheet focuses on describing data through measuring its central tendency and variability. These measurements will give us an idea of what

### PROPERTIES OF MEAN, MEDIAN

PROPERTIES OF MEAN, MEDIAN In the last class quantitative and numerical variables bar charts, histograms(in recitation) Mean, Median Suppose the data set is {30, 40, 60, 80, 90, 120} X = 70, median = 70

### 6. Methods 6.8. Methods related to outputs, Introduction

6. Methods 6.8. Methods related to outputs, Introduction In order to present the outcomes of statistical data collections to the users in a manner most users can easily understand, a variety of statistical

### Data Analysis: Describing Data - Descriptive Statistics

WHAT IT IS Return to Table of ontents Descriptive statistics include the numbers, tables, charts, and graphs used to describe, organize, summarize, and present raw data. Descriptive statistics are most

### Statistics Summary (prepared by Xuan (Tappy) He)

Statistics Summary (prepared by Xuan (Tappy) He) Statistics is the practice of collecting and analyzing data. The analysis of statistics is important for decision making in events where there are uncertainties.

### Using SPSS, Chapter 2: Descriptive Statistics

1 Using SPSS, Chapter 2: Descriptive Statistics Chapters 2.1 & 2.2 Descriptive Statistics 2 Mean, Standard Deviation, Variance, Range, Minimum, Maximum 2 Mean, Median, Mode, Standard Deviation, Variance,

### Numerical Measures of Central Tendency

Numerical Measures of Central Tendency Often, it is useful to have special numbers which summarize characteristics of a data set These numbers are called descriptive statistics or summary statistics. A

### Descriptive Statistics

Descriptive Statistics Suppose following data have been collected (heights of 99 five-year-old boys) 117.9 11.2 112.9 115.9 18. 14.6 17.1 117.9 111.8 16.3 111. 1.4 112.1 19.2 11. 15.4 99.4 11.1 13.3 16.9

### Mathematics. Probability and Statistics Curriculum Guide. Revised 2010

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

### DESCRIPTIVE STATISTICS

CHAPTER 1 DESCRIPTIVE STATISTICS Page Contents 1.1 Introduction 1. Some Basic Definitions 1.3 Method of Data Collection 3 1.4 Primary and Secondary Data 6 1.5 Graphical Descriptions of Data 7 1.6 Frequency

### Content DESCRIPTIVE STATISTICS. Data & Statistic. Statistics. Example: DATA VS. STATISTIC VS. STATISTICS

Content DESCRIPTIVE STATISTICS Dr Najib Majdi bin Yaacob MD, MPH, DrPH (Epidemiology) USM Unit of Biostatistics & Research Methodology School of Medical Sciences Universiti Sains Malaysia. Introduction

### Describe what is meant by a placebo Contrast the double-blind procedure with the single-blind procedure Review the structure for organizing a memo

Readings: Ha and Ha Textbook - Chapters 1 8 Appendix D & E (online) Plous - Chapters 10, 11, 12 and 14 Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability

### Lecture 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

### ( ) ( ) Central Tendency. Central Tendency

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

### STATS8: 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

### TOPIC. Directorate of Distance Education (DDE) Program: P.G. Subject: Geography. Class: Previous. Paper: V(a)

TOPIC Measures Dispersions: Range, Quartile Deviation, Mean Deviation, Standard Deviation and Lorenz Curve-Their Merits and Limitations and Specific Uses Prepared By: Dr. Kanhaiya Lal, Assistant Pressor,

### Pie 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

### Biyani's Think Tank. Concept based material. Business Statistics. BBA lll Sem

Biyani's Think Tank Concept based material Business Statistics BBA lll Sem Dr. Archana Dusad Assistant Professor Deptt. of Commerce & Management Biyani Girls College, Jaipur 2 Published by : Think Tanks

### Class B.Com. III Sem.

SYLLABUS Class B.Com. III Sem. Subject Statistics UIT I UIT II UIT III Meaning, definition, significance, scope and limitations of statistical investigation process of data collection, primary and secondary

### Chapter 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)

### Descriptive 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

### IB Mathematics SL :: Checklist. Topic 1 Algebra The aim of this topic is to introduce students to some basic algebraic concepts and applications.

Topic 1 Algebra The aim of this topic is to introduce students to some basic algebraic concepts and applications. Check Topic 1 Book Arithmetic sequences and series; Sum of finite arithmetic series; Geometric

### Interactive Math Glossary Terms and Definitions

Terms and Definitions Absolute Value the magnitude of a number, or the distance from 0 on a real number line Additive Property of Area the process of finding an the area of a shape by totaling the areas

### Statistical Foundations: Measures of Location and Central Tendency and Summation and Expectation

Statistical Foundations: and Central Tendency and and Lecture 4 September 5, 2006 Psychology 790 Lecture #4-9/05/2006 Slide 1 of 26 Today s Lecture Today s Lecture Where this Fits central tendency/location

### Statistics. 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

### III. GRAPHICAL METHODS

Pie Charts and Bar Charts: III. GRAPHICAL METHODS Pie charts and bar charts are used for depicting frequencies or relative frequencies. We compare examples of each using the same data. Sources: AT&T (1961)

### Lecture 3: Measure of Central Tendency

Lecture 3: Measure of Central Tendency Donglei Du (ddu@unb.edu) Faculty of Business Administration, University of New Brunswick, NB Canada Fredericton E3B 9Y2 Donglei Du (UNB) ADM 2623: Business Statistics

### HISTOGRAMS, 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

### Dr. Peter Tröger Hasso Plattner Institute, University of Potsdam. Software Profiling Seminar, Statistics 101

Dr. Peter Tröger Hasso Plattner Institute, University of Potsdam Software Profiling Seminar, 2013 Statistics 101 Descriptive Statistics Population Object Object Object Sample numerical description Object