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STATISTICS MADE EASYPosted by emerjoytvale in Education on August 12th, 2010 THE STUDY OF STATISTICS WHY STATISTICS? · important in empirical studies · aids in decision making · helps to forecast or predict future outcomes · estimates unknown values · aids in making inferences, comparison or establishing relationship · summarizes data DEFINITIONS · (plural sense) – a set of numerical data or observations Ex. o Vital statistics in a beauty pageant o Yearly income o Monthly expenses o First semester grades · (singular sense) – a branch of science which deals with the collection, presentation, analysis and interpretation of data. MAIN DIVISION OF STATISTICS · Descriptive Statistics – pertains to the methods dealing with the collection, organization and analysis of a set of data without making conclusions, predictions or inferences about a larger set. o
the main goal is simply to
provide a description of a particular data set o
the conclusions or the
important characteristics apply only to the data on hand · Inferential Statistic – pertains to the methods dealing with making inferences, estimation or prediction about a larger set of data using the information gathered from a subset of this larger set o
the main goal is not merely to
provide a description of a particular data set but also to make prediction and inferences
based on the available information gathered o
the conclusions or the
important characteristics apply to a larger set from which the data on hand is
only a subset Collection of Data – refers to the method of data gathering Presentation of Data – refers to the process of organizing data such as tabulation, presenting through the use of charts, graph or paragraphs Analysis of Data – refers to the methods of obtaining necessary, relevant and noteworthy information from the given data Interpretation of Data – refers to the tools of drawing of conclusions or inferences from the analyzed data Examples: Descriptive Statistics · A college dean wants to determine the average semestral enrolment in the past 5 school years. · An instructor wants to know the exact number of students who pass in his subject. · A school president wants to determine the number of student-dropouts for this current school year Inferential
Statistics · A college dean wants to forecast the average semestral enrolment based on the enrolment for the last 5 school years · An instructor would like to predict the number of students who will pass in his subject based on the number of failures last year · A school president would like to estimate the number of student dropouts next school year based on the current data. DESCRIPTIVE STATISTICS POPULATION AND SAMPLE · Population – collection of all cases in which the researcher is interested in a statistical study - the entity that the researcher wished to understand Examples: · All subjects of University of Bohol · All department heads in UB · Sample – a portion or a subset of the population from which the information is gathered - a representation of the population Examples: · All students of University of Bohol coming from the rural areas · All department heads in UB who have finished Ph.D. Degree · Parameter – a numerical characteristic of a population - denoted by small Greek letters Examples: · µ - population mean · σ – population standard deviation · Statistic – a numerical characteristic of a sample - Denoted by lower case letters of the English alphabet Examples: · X – sample mean · SD – sample standard deviation TYPES OF DATA · Variable – a characteristic or attribute of persons or objects which assumes different values or label · Measurement – process of assigning the value or label of a particular variable for a particular experimental unit · Experimental unit – the person or the object on which a variable is measured · Classification of Variables Ø Qualitative Variable – yields categorical or qualitative responses Examples: Civil Status (Single, Married, Widow, etc.) Religious Affiliation (Catholic, Protestant, etc.) Ø Quantitative Variable – yields numerical responses representing an amount or quantity Examples: Height, Weight, no. of children Types of Quantitative Variable Ø Discrete Variable – assumes finite or countable infinite values such as 1,2,3,… Example: no. of children (0,1,2,3,…) no. of student-dropouts Ø Continuous Variable – cannot take on finite values but the values are related/associated with points on an interval of the real line Examples: Height (5’4”) Weight (130.42 kilos) Temperature (32.5°C) Levels of Measurement Ø NOMINAL LEVEL - Crudest form of measurement - Numbers or symbols are used for the purpose of categorizing subjects into groups - The categories are mutually exclusive, that is being in one category automatically excludes inclusion in another - The categories are exhaustive, that is all possible categories of a variable should be included Examples: Sex: 1 – Male 0 – Female Faculty Tenure: 1 – Tenured 0 – Non-Tenured Ø ORDINAL LEVEL - Improvement of nominal level - Order/rank the data in a somewhat “bottom to top” or “low to high” manner Examples: Class Standing (Excellent, Good, Poor) Teacher Evaluation 1 – Poor 2 – Fair 3 – Good 4 – Very Good Ø INTERVAL LEVEL - Possesses the properties of the nominal and ordinal levels - Distances between any two numbers on the scale are known - Does not have a stable starting point (an absolute zero) Example: Consider the IQ scores of four students 70, 140, 75 and 145 Here, we can say that the difference between 70 and 140 is the same as the differences between 75 and 145 but we cannot claim that the second student is twice as intelligent as the first. Ø RATIO LEVEL - Possesses all the properties of the nominal, ordinal and interval levels and in addition, this has or absolute zero point - We can classify it, place it in proper order - We can also compare magnitudes Examples: Age, Income, Exam Scores SUMMARY CHARACTERISTICS OF LEVELS OF MEASUREMENT Levels of Measurement Classify Order Equal Limits Absolute Zero NOMINAL Yes No No No ORDINAL Yes Yes No No INTERVAL Yes Yes Yes No RATIO Yes Yes Yes Yes · Other Classification of Data Ø Raw Data - data in their original form and structure Ø Grouped Data - data placed in tabular form Ø Primary Data - measured and gathered by the researcher that published it Ø Secondary Data - any republication of data by another researcher or agency METHODS OF DATA COLLECTION ·
Observation Method - Data can be obtained by observing the behavior of persons or objects but only at a particular time of occurrence - The data obtained is called an observational data ·
Experimental Method - Especially useful when one wants to collect data for cause and effects studies - There is actual human interference with the conditions or situations that can affect the variable under study - Prevalent in scientific researches ·
Use of Existing Studies - CHED or DECS enrollment data - Census Data ·
Registration Method - Respondents provide the necessary information in observance and compliance with existing laws - Our registration, birth registration, student registration, voter’s registration ·
Survey Method - The desired information is obtained through asking questions Common Forms of Survey Method Ø Personal Interview
Method - There is a person-to-person contact between the interviewer and the interviewee - Considered as one of the most effective methods of data collection because accurate and precise information can be directly obtained and verified from the respondents - Higher response rate - Can be administered to the respondents one at a time Ø Questionnaire Method - Considered the easiest method of data collection - Utilizes an instrument which is the questionnaire as a tool - Lower response rate - Can be administered to a large number of respondents simultaneously General Classification of Data Collection Ø Census or Complete
Enumeration - Method of gathering data from every unit in the population - Not always possible because of money, time and effort Ø Survey Sampling - Method of gathering data from every unit in the selected sample - Reduces cost, greater speed, scope and accuracy PROBABILITY AND NON-PROBABILITY SAMPLING · Probability Sampling – sampling procedure in which every element in the population has known non-zero chance of being included in the sample Common Methods of Probability Sampling · Simple Random Sampling (SRS) - Sampling procedure in which every element in the population has an equal chance of being included in the sample - Select n units out of N units in the population - The selection is through lottery or the use of the table of random numbers ·
Stratified Random Sampling - The population of N units into subpopulations called strata which consists of more or less homogeneous units - Perform simple random from each strata, the selection of which is independent in different strata - Requires a so-called “stratification variable” Example: Consider a population consisting of all UB students § Stratify the population by colleges (stratification variable)
§ Stratify the population by year level (stratification variable).
·
Systematic Sampling - Done with a random start - Select the sample by taking every k^{th} unit from an ordered population - k is called the sampling interval and 1/k is the sampling fraction Example: Suppose we select n=12 students from a population of N=50. To employ systematic sampling, divide N by n to get k, that is K = N/n = 50/12 = 4.667 ≈ 5
We chose every 5^{th} unit. Thus if the random start r = 9^{th} unit, then the sample comprise of students number 9, 14, 19, 24, 29, 34, 39, 44, 49 and 4. · Cluster Sampling - A variation of stratified sampling where the strata correspond to clusters, however with heterogeneous characteristics or attributes. Example: A college unit may be considered as one cluster. · Multi-stage Sampling - A sampling procedure wherein the population is divided into a sequence of sampling units corresponding to the different sampling stages. Common Methods of Non-Probability Sampling ·
Purposive Sampling ·
Quota Sampling ·
Convenience Sampling PRESENTATION OF DATA Data can be represented in various modes such as the textual, tabular and graphical displays. ·
Textual - Here, the data are presented by use of texts, phrases or paragraphs - Very common among newspaper stories, depicting only the salient findings ·
Tabular - This is a more reliable and effective way of showing relationships or comparisons of data through the use of tables - In many cases, the tables are accompanied by a short narrative explanation to make the facts clearer and more understandable ·
Graphical - Most effective way of presenting statistical findings by use of statistical graphs - Attracts attention to the reader - Sets the tone for clearer interpretation of findings especially in making comparisons Common Form of Graphs · Bar Graph – uses rectangular bars the length of which represents the quantity or frequency for each type category Example: We are given the ff. enrolment data of XYZ College for the Academic Year 1995-1996
We can draw the bar graph by placing the year levels on the horizontal axis and the frequency on the vertical axis as follows: e n r o l m e n t
· Multiple Bar Graph – useful when the researcher wants to compare figures – uses legend to guide the viewer in analyzing the data Example: For the same data set, the multiple bar graph is shown below. · Pie Chart - used to present quantities that make up a whole - the slices of the pie are drawn in proportion to the different values each class, item, group or category - the total area of the pie is 100% · Line Chart – especially useful in showing trends over a period of time Example: The following data shows the number of student-dropouts of UB from 1995-1999
The line chart of the above data set is shown below: THE FREQUENCY DISTRIBUTION
TABLE Data in its original form and structure are called raw data. Example: Consider the following final exam scores of 40 students
When these scores are arranged either in ascending or descending magnitude, then such an arrangement is called an array. It is usually helpful to put the raw data in an array because it is easy to identify the extreme values or the values where the scores most cluster. When these data are placed into a system wherein they are organized, then these partake the nature of grouped data. This procedure of organizing data into groups is called a frequency distribution table (FDT). Example: The following presents a frequency distribution table of the scores of seventy-five Statistics students.
Some Basic Terms · Class Interval - the numbers defining the class - consists of the end numbers called the class limits namely the upper limit and the lower limit · Class Frequency - shows the number of observation falling in the class · Class Boundaries - these are true class limits - LOWER CLASS BOUNDARY (LCB) is defined as the middle value of the lower class limits of the class and the upper class boundary of the preceding class - UPPER CLASS BOUNDARY (UCB) is the middle value between the upper class limits of the class and the lower limit of the next class · Class size - the difference between the upper limits of the class and the preceding class · Class Mark - midpoint of a class interval · Open-ended class - one which has no lower limit or upper limit · Cumulative frequency - shows the accumulated frequencies of successive classes - GREATER
THAN CF ( - LESS
THAN CF ( Example:
Steps in Constructing a
Frequency Distribution Table (FDT) Step 1: Determine the number of classes. For first approximation, it is suggested to use the STURGES APPROXIMATION FORMULA. K = 1 + 3.322 log n where K = approximate number of classes n = number of cases Step 2: Determine the ranges R. R = maximum value – minimum value Step 3: Determine the approximate size, i using the formula. i = R/K It is usually convenient to round off i to a convenient number. Step 4: Determine the lowest class limit (or the first class). This class should include the minimum value in the data set. Step 5: Determine all class limits by adding the class size i to the limits of the previous class. Step 6: Tally the scores/observations falling in each class. Example: Constructing the FDT of the final exam scores of 40 students. Step 1: K = 1 + 3.322 log n = 1 + 3.322 log (40) = 1 + 3.322 (1.60205) = 1 + 5.32204 = 6.3220 K = 6 Step 2: R = max – min = 93 – 32 R = 61 Step 3: i = R/K = 61/6 = 10.167 i = 10 Step 4: Let us decide to start at the minimum value. Thus, the lowest class is the class 32 – 41 Step 5: The classes are constructed by adding 10 to each class limit 32 – 41 42 – 51 52 – 61 62 – 71 72 – 81 82 – 91 92 – 101 Step 6: Determine now the frequency of each class by tallying the scores falling in each class.
We now proceed constructing the complete frequency distribution table.
Graphs Associated with the Frequency Distribution Table · Histogram – a bar graph in which the class boundaries are plotted (on the horizontal axis) against the class frequencies (on the vertical axis) The following depicts the histogram of the frequency distribution table of the final exam scores of 40 students. Frequency Class Boundaries · Frequency Polygon – a line chart that is constructed by plotting the class marks against the class frequencies. – The graph is obtained by connecting the consecutive points by use of straight lines. – The polygon is closed by adding additional classmarks at each end with a frequency of zero. Frequency Class Marks (Scores) · Ogives – graphs associated with cumulative frequencies Cumulative Frequency Class Boundaries (Class Scores) ·
Ogive – graph where the > CF
is plotted against the LCB Cumulative Frequency Class
Boundaries (Class Scores) MEASURES OF CENTRAL
TENDENCY Measures of Central Tendency · Show the centrality of the data · Measures of the average · Common measures are the mean, the median and the mode THE MEAN Ø For Ungrouped Data · Population Mean (µ) _________ N ·
_________ n Example: The following are scores of 10 sample students: 43, 32, 72, 31, 28 25, 45, 38, 42, 38
43 + 32 + 72 + 31 + 28 + 25 + 45 + 38 + 42 + 38 394 10 10 Ø Approximate the mean for Grouped Data _________ n f_{i }= frequency of the i^{th} class x_{i} = midpoint of the i^{th} class n = number of cases Example: Consider the FDT scores of 75 Statistics students
Thus, _________ n
= 2787.5 / 75 Properties of the Mean · The mean is the most widely used measure, which applies only to interval/ratio data. · It is affected by extreme values. · Since it is a calculated average, and its value is determined in every observation, then the mean may not be the actual value or number in the data set. · The sum of the deviations about the mean is zero. · If a constant k is added (or subtracted) to every observation in the data set, then the mean of the new data set increases (or decreases) by the same constant k. · If a constant k is multiplied to every observation in the data set, the mean of the new data set is a constant multiple of the original mean. THE MEDIAN Ø For Ungrouped Data - Denoted by Me - The middle most value when the observation are arranged either in ascending or descending order - If a data set has an even number of observations, then the median is the average (the Mean) of the two most middle values. Examples: a.) Consider the ff. set of scores 12, 34, 45, 72, 38, 49, 65 Putting the scores in an array; 12, 34, 38, 45, 49, 65, 72 Observe that Me = 45 b.) Consider the next ff. set of score 8, 16, 24, 7, 21, 17, 19, 18, 5, 26 Putting the scores in an array; 5, 7, 8, 16, 17, 18, 19, 21, 24, 26 Observe that the two middle most values are 17 and 18. Thus, the median is 17+18 35 Me = ---------- = ---- = 17.5 2 2 Ø Approximating the median for Grouped Data where: LCB_{me} = lower class boundary of the median class c = class size n = number of cases f_{me} = frequency of the median class median class = class where the Example: Consider again the following FDT:
To approximate the median, we construct first the
Then we locate next
the median class. As defined, the median class is the class where the Thus, LCB_{me} = 29.5 i = 10 n/2 = 37.5 f_{me }= 23 Substituting these values to the formula, we obtain:
Properties of the Median
THE MODE Ø For Ungrouped Data · Denoted by Mo · Value in the data set has the highest frequency Example: Consider the ff. data set whose values are IQ scores Data Set A : 77, 83, 91, 85, 83, 100 Data Set B : 88, 92, 71, 88, 71, 36 Data Set C : 96, 43, 79, 68, 83, 110 Notice that the mode for Data Set A is 83; the modes for Data Set B are 88 and 71 and we say that this data set is bi-modal. Data Set C does not have a mode. Ø Approximating the Mode for Grouped Data a. b. where: LCB_{me} = lower class boundary of the median class i = class size f_{me} = frequency of the modal class f_{i} = frequency of the class preceding the modal class f_{2 }= frequency of the class following the modal class Example: Consider again the following FDT:
To approximate the mode, let us first locate the modal class, the class which has the highest frequency. Thus, 30 – 39 is the modal class and LCB_{me} = 29.5 i = 10 f_{me} = 23 f_{i} = 14 f_{2 }= 22 Substituting the values we obtain:
a. b.
For larger cases, n ≥ 100 a.
Mo = 3Me - 2 b.
Mo = Other Kinds of Mean I. Weighted Mean (WM) = ∑xw / N II. Geometric Mean, (GM) · Used to derived the average of indexes, relative values and percentages · n^{th} root of the product of n number of values · antilog of the logarithms of the middle values multiplied by the frequencies divided by the total frequency distribution = (X_{1 }∙ X_{2 }∙ X_{3} ∙ ∙ ∙X_{n})^{1/n} GM = III. Harmonic Mean, (HM) · Used for spatial measurements, lengths, areas and volumes. · The reciprocal of the arithmetic mean of the reciprocals of the values. HM = HM = Example: 10, 15, 8, 10, 13, 12, 10, 14
1. GM = (10 ∙ 5 ∙ 8 ∙ 10 ∙ 13 ∙ 12 ∙ 10 ∙ 14)1/8 = (262080000)^{1/8} = 11.28 GM =
2.
HM = =
=
=
11.06 MEASURES OF DISPERSION OR
VARIABILITY ·
Measure of Dispersion – indicate how the
data are dispersed or scattered about the average ·
Classifications: o
Measures of Absolute Dispersion -
Expressed in the original units
of the original observations THE RANGE -
Difference between the largest
and the smallest values -
Maximum value minus minimum
value -
For grouped data, the range is
defined as the difference between the upper class limit of the highest class
and the lower class limit of the lowest class -
Simplest/roughest measure Example: Consider the FDT
The Range R = 59 – 10 = 49 (for discrete) R = 59.5 – 9.5 = 50 (for continuous) Properties of the Range · It is a weak measure because it tells only the extreme values and does not provide information on the values between · It is greatly affected by outliers · For open-ended frequency distributions, the range cannot be computed THE STANDARD DEVIATION Ø For Population Ø For Sample Example: Consider the following scores of 5 students taken as samples: 8, 6, 3, 4, 4 Then 5 5 Thus,
The Standard deviation is therefore: S = S = S = S = Computational Formula for the Sample Standard Deviation for Ungrouped Data Example: To verify the computed standard deviation in the previous sample, we have
Thus, s = 2 Ø Approximating the Standard Deviation for Grouped Data where f_{1} = frequency of the i^{th} class x_{1} = classmark of the i^{th }class x = mean of the frequency distribution n = number of cases Example: Consider again the FDT:
The mean of this FDT was computed to be x = 37.17. The standard deviation is computed as follows:
∑f_{1}(x_{1}- Hence, Ø Computational Formula for Approximating the Standard Deviation for Grouped Data Properties of Standard
Deviation · Since it is a function of the mean, the standard deviation is affected by every value of the data set. Thus, it is sensitive against the presence of few extreme values. · If each observation of a data set is added or subtracted by the same amount k, then the standard deviation of the new data set is the same as the standard deviation of the original data set. · If each of the data set is multiplied by a constant k, then the standard deviation of the new data set is equal to k times the standard deviation of the standard deviation of the original data set. OTHER MEASURES OF
DISPERSION · Coefficient of Variation (CV) - Ratio of the standard deviation to its mean - Especially useful when one compares the variability of one data set with another data set having different units CV = s/ · Index of Qualitative Variation (IQV) - Dispersion measures for qualitative nominal or ordinal variable - If all the values f a variable are in one category, then there is no variation and IQV = 0 - If all the values are distributed evenly across the categories, IQV = 1, maximum value IQV = k(N^{2} - ∑f^{2}) / N^{2} (k-1) where k = number of categories N = number of observations MEASURES OF SKEWNESS - Shows the degree of asymmetry or departure from symmetry of a distribution - Indicates also the direction of skewness § Types of Skewness o Positively Skewed § Longer tail to the right § § This happens when Mo
Me Example: If a teacher gives a very hard exam, then one can expect that the distribution of scores will be positively skewed. o Negatively Skewed § Longer tail to the left § § This happens when Example: A very easy exam will result to a distribution of scores which is negatively skewed. Ø Pearson’s Coefficients of Skewness where Mo = mode Me = median SD = standard deviation We prefer to use Formula 2 over Formula 1 in as much as the mode does not always exist. Example: Recall the Statistics obtained from the FDT of scores of fifteen statistics students. Me = 37.54 s = 11.31 Thus, we say that the distribution of scores is skewed to the left. MEASURES OF KURTOSIS Ø Describes the relative flatness or peakness of a distribution · Platykurtic – relatively flat ·
·
Mesokurtic – is between the platykurtic
and leptokurtic curves; approaches or look like the “normal curve” Ø Pearson’s Coefficient of Kurtosis where x_{1} = observations
in a data set x = the
sample mean n = no.
of cases SD = standard
deviation Observe that if: Ku < 3, then the distribution is
platykurtic Ku = 3, then the distribution is mesokurtic Ku > 3, then the distribution is leptokurtic compilation of Dr. Libot Also See: Yes Yes, Student Dropouts, Probability Sampling, Frequency Distribution, Yes, Year, Variable More By This AuthorMake Your Academic Performane Highly Noticeable BEST RESEARCH FORMAT IN TOWN Brain Fever FLU: Critical or Not Roles of Nurses in Health Assessment Related Articles
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