Animals Arts Business Computers Education Finance Games Health Home Internet News Other Recreation Science Shopping Society Sports Technology | ## STATISTICS MADE EASYPosted by emerjoytvale in Society on August 12th, 2010
WHY STATISTICS? · important in empirical studies · aids in decision making · helps to forecast or predictfuture outcomes · estimates unknown values · aids in making inferences,comparison or establishing relationship · summarizes data DEFINITIONS · (plural sense) – a set ofnumerical data or observations Ex. o Vital statistics in a beautypageant o Yearly income o Monthly expenses o First semester grades · (singular sense) – a branch ofscience which deals with the MAIN DIVISION OFSTATISTICS · o the main goal is simply toprovide a description of a particular data set o the conclusions or theimportant characteristics apply only to the data on hand · o the main goal is not merely toprovide a description of a particular data set but also to make prediction and inferencesbased on the available information gathered o the conclusions or theimportant characteristics apply to a larger set from which the data on hand isonly a subset Collection of Data – refers to the method of data gathering Presentationof Data – refers to the process of organizing data suchas tabulation, presenting through the use of charts, graph or paragraphs Analysisof Data – refers to the methods of obtaining necessary,relevant and noteworthy information from the given data Interpretationof Data – refers to the tools of drawing of conclusionsor inferences from the analyzed data Examples:
· A college dean wants todetermine the average semestral enrolment in the past 5 school years. · An instructor wants to know theexact number of students who pass in his subject. · A school president wants todetermine the number of student-dropouts for this current school year
· A college dean wants toforecast the average semestral enrolment based on the enrolment for the last 5school years · An instructor would like topredict the number of students who will pass in his subject based on the numberof failures last year · A school president would liketo estimate the number of student dropouts next school year based on thecurrent data. DESCRIPTIVE STATISTICS POPULATION AND SAMPLE · Population – collection of allcases in which the researcher is interested in a statistical study - the entity that the researcherwished to understand Examples: · All subjects of University ofBohol · All department heads in UB · Sample – a portion or a subset ofthe population from which the information is gathered - a representation of thepopulation Examples: · All students of University ofBohol coming from the rural areas · All department heads in UB whohave finished Ph.D. Degree · Parameter – a numerical characteristicof a population - denoted by small Greek letters Examples: · µ - population mean · σ – population standard deviation · Statistic – a numericalcharacteristic of a sample - Denoted by lower case lettersof the English alphabet Examples: · X – sample mean · SD – sample standard deviation TYPESOF DATA · Variable – a characteristic orattribute of persons or objects which assumes different values or label · Measurement – process ofassigning the value or label of a particular variable for a particularexperimental unit · Experimental unit – the person orthe object on which a variable is measured · Classification of Variables Ø Qualitative Variable – yieldscategorical or qualitative responses
Ø Quantitative Variable – yieldsnumerical responses representing an amount or quantity
Typesof Quantitative Variable Ø Discrete Variable – assumesfinite or countable infinite values such as 1,2,3,…
Ø Continuous Variable – cannot takeon finite values but the values are related/associated with points on aninterval of the real line
Temperature(32.5°C) Levelsof Measurement Ø NOMINAL LEVEL - Crudest form of measurement - Numbers or symbols are used forthe purpose of categorizing subjects into groups - The categories are mutuallyexclusive, that is being in one category automatically excludes inclusion inanother - The categories are exhaustive,that is all possible categories of a variable should be included
Faculty Tenure: 1 – Tenured 0 – Non-Tenured Ø ORDINAL LEVEL - Improvement of nominal level - Order/rank the data in asomewhat “bottom to top” or “low to high” manner
Teacher Evaluation 1 – Poor 2– Fair 3– Good 4– Very Good Ø INTERVAL LEVEL - Possesses the properties of thenominal and ordinal levels - Distances between any twonumbers on the scale are known - Does not have a stable startingpoint (an absolute zero)
Here,we can say that the difference between 70 and 140 is the same as thedifferences between 75 and 145 but we cannot claim that the second student istwice as intelligent as the first. Ø RATIO LEVEL - Possesses all the properties ofthe nominal, ordinal and interval levels and in addition, this has or absolutezero point - We can classify it, place it inproper order - We can also compare magnitudes
SUMMARY CHARACTERISTICS OF LEVELS OF MEASUREMENT Levels of Measurement Classify Order EqualLimits 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 Ø Ø Ø METHODSOF DATA COLLECTION · Observation Method - Data can be obtained byobserving the behavior of persons or objects but only at a particular time ofoccurrence - The data obtained is called anobservational data · Experimental Method - Especially useful when onewants to collect data for cause and effects studies - There is actual humaninterference with the conditions or situations that can affect the variableunder study - Prevalent in scientific researches · Use of Existing Studies - CHED or DECS enrollment data - Census Data · Registration Method - Respondents provide thenecessary information in observance and compliance with existing laws - Our registration, birthregistration, student registration, voter’s registration · Survey Method - The desired information isobtained through asking questions CommonForms of Survey Method Ø Personal InterviewMethod - There is a person-to-personcontact between the interviewer and the interviewee - Considered as one of the mosteffective methods of data collection because accurate and precise informationcan be directly obtained and verified from the respondents - Higher response rate - Can be administered to therespondents one at a time Ø Questionnaire Method - Considered the easiest methodof data collection - Utilizes an instrument which isthe questionnaire as a tool - Lower response rate - Can be administered to a largenumber of respondents simultaneously GeneralClassification of Data Collection Ø Census or CompleteEnumeration - Method of gathering data fromevery unit in the population - Not always possible because ofmoney, time and effort Ø Survey Sampling - Method of gathering data fromevery unit in the selected sample - Reduces cost, greater speed,scope and accuracy PROBABILITYAND NON-PROBABILITY SAMPLING · Probability Sampling – samplingprocedure in which every element in the population has known non-zero chance ofbeing included in the sample CommonMethods of Probability Sampling · Simple Random Sampling (SRS) - Sampling procedure in whichevery element in the population has an - Select n units out of N unitsin the population - The selection is throughlottery or the use of the table of random numbers · Stratified Random Sampling - The population of N units intosubpopulations called strata which consists of more or less homogeneous units - Perform simple random from eachstrata, the selection of which is independent in different strata - Requires a so-called “stratification variable”
§ Stratify the population by
§ Stratify the population by
· Systematic Sampling - Done with a random start - Select the sample by takingevery k - k is called the samplinginterval and 1/k is the sampling fraction
K = N/n = 50/12 = 4.667 ≈ 5
We chose every 5 9, 14, 19, 24, 29, 34, 39, 44, 49 and 4. · Cluster Sampling - A variation of stratified samplingwhere the strata correspond to clusters, however with heterogeneouscharacteristics or attributes.
· Multi-stage Sampling - A sampling procedure whereinthe population is divided into a sequence of sampling units corresponding tothe different sampling stages.
· Purposive Sampling · Quota Sampling · Convenience Sampling PRESENTATION OF DATA Data can be represented in variousmodes such as the textual, tabular and graphical displays. · Textual - Here, the data are presented byuse of texts, phrases or paragraphs - Very common among newspaperstories, depicting only the salient findings · Tabular - This is a more reliable andeffective way of showing relationships or comparisons of data through the useof tables - In many cases, the tables areaccompanied by a short narrative explanation to make the facts clearer and moreunderstandable · Graphical - Most effective way ofpresenting statistical findings by use of statistical graphs - Attracts attention to thereader - Sets the tone for clearerinterpretation of findings especially in making comparisons CommonForm of Graphs · Bar Graph – uses rectangular barsthe length of which represents the quantity or frequency for each type category
We can draw the bar graph by placingthe year levels on the horizontal axis and the frequency on the vertical axisas follows: e n r o l m e n t
· Multiple Bar Graph – useful when the researcher wants tocompare figures – uses legend to guide the viewer inanalyzing the data
· Pie Chart - used to present quantities that makeup 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 inshowing trends over a period of time
Theline chart of the above data set is shown below:
Datain its original form and structure are called raw data.
When these scores are arrangedeither in ascending or descending magnitude, then such an arrangement is calledan array. It isusually helpful to put the raw data in an array because it is easy to identifythe extreme values or the values where the scores most cluster. When these data are placed into asystem wherein they are organized, then these partake the nature of groupeddata. This procedure of organizing data into groups is called a frequency distribution table (FDT).
· Class Interval - the numbers defining the class - consists of the end numberscalled the class limits namely the upper limit and the lower limit · Class Frequency - shows the number of observationfalling in the class · Class Boundaries - these are true class limits -LOWER CLASS BOUNDARY (LCB) is defined as the middle value of the lower classlimits of the class and the upper class boundary of the preceding class -UPPER CLASS BOUNDARY (UCB) is the middle value between the upper class limitsof the class and the lower limit of the next class · Class size - the difference between the upper limits of theclass and the preceding class · Class Mark - midpoint of a class interval · Open-ended class - one which has no lower limit or upperlimit · Cumulative frequency - shows the accumulated frequencies ofsuccessive classes - GREATERTHAN CF ( - LESSTHAN CF (
Step 1: Determine the number of classes. For first approximation, itis 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, It isusually convenient to round off Step 4: Determine the lowest class limit (or the first class). Thisclass should include the minimum value in the data set. Step 5: Determine all class limits by adding the class size Step 6: Tally the scores/observations falling in each class.
Step 1: K = 1+ 3.322 log n = 1 + 3.322 log (40) = 1 + 3.322 (1.60205) = 1 + 5.32204 = 6.3220
Step 2: R = max– min = 93 – 32 Step 3: = 61/6 = 10.167 Step 4: Let us decide to start at the minimum value. Thus, thelowest 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 thescores falling in each class.
We now proceed constructing the complete frequency distributiontable.
· Histogram – a bar graph inwhich the class boundaries are plotted (on the horizontal axis) against theclass frequencies (on the vertical axis) The followingdepicts the histogram of the frequency distribution table of the final exam scoresof 40 students. Frequency Class Boundaries · Frequency Polygon – a line chart that is constructed byplotting the class marks against the class frequencies. – The graph is obtained by connecting the consecutive points by useof straight lines. – The polygon is closed by adding additional classmarks at each endwith a frequency of zero. Frequency Class Marks (Scores) · Ogives – graphs associated withcumulative frequencies Cumulative Frequency Class Boundaries (Class Scores) · Ogive – graph where the > CFis plotted against the LCB Cumulative Frequency ClassBoundaries (Class Scores)
· Show the centrality of the data · Measures of the average · Common measures are the mean,the median and the mode
Ø For Ungrouped Data · Population Mean ( _________ N · _________ n
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 forGrouped Data _________ n f x n = numberof cases
Thus, _________ n
= 2787.5 / 75
· The mean is the most widelyused measure, which applies only to interval/ratio data. · It is affected by extremevalues. · Since it is a calculatedaverage, and its value is determined in every observation, then the mean maynot be the actual value or number in the data set. · The sum of the deviations aboutthe mean is zero. · If a constant k is added (orsubtracted) to every observation in the data set, then the mean of the new dataset increases (or decreases) by the same constant k. · If a constant k is multipliedto every observation in the data set, the mean of the new data set is aconstant multiple of the original mean.
Ø For Ungrouped Data - Denoted by Me - The middle most value when theobservation are arranged either in ascending or descending order - If a data set has an evennumber of observations, then the median is the average (the Mean) of the twomost middle values.
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 ofscore 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, themedian is 17+18 35 Me = ---------- = ---- = 17.5 2 2 Ø Approximating the median forGrouped Data where: LCB c = class size n = number of cases
f median class = class where the
To approximate the median, we construct first the
Then we locate nextthe median class. As defined, the median class is the class where the Thus, LCB n/2 = 37.5 f Substituting these values to the formula, we obtain:
- The median is an ordinal and positional measure
- Not affected by extreme values compared to the mean
Ø For Ungrouped Data · Denoted by Mo · Value in the data set has thehighest frequency
DataSet A : 77, 83, 91, 85, 83, 100 DataSet B : 88, 92, 71, 88, 71, 36 DataSet C : 96, 43, 79, 68, 83, 110 Notice that themode for Data Set A is 83; the modes for Data Set B are 88 and 71 and we saythat 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
f f f
To approximate themode, let us first locate the modal class, the class which has the highestfrequency. Thus, 30 – 39 is the modal class and LCB
f f f Substitutingthe values we obtain: - Crude Mode – rough
a. b. - Refined Mode
For larger cases, n ≥ 100 a. Mo = 3Me - 2 b. Mo = OtherKinds of Mean I. Weighted Mean (WM) = ∑xw / N II. Geometric Mean, (GM) · Used to derived the average ofindexes, relative values and percentages · n · antilog of the logarithms ofthe middle values multiplied by the frequencies divided by the total frequencydistribution =(X GM = III. Harmonic Mean, (HM) · Used for spatial measurements,lengths, areas and volumes. · The reciprocal of thearithmetic mean of the reciprocals of the values. HM = HM =
1. GM = (10 ∙ 5 ∙ 8 ∙ 10 ∙ 13 ∙ 12 ∙ 10 ∙ 14)1/8 =(262080000) =11.28 GM = 2. HM = =
= =
· · Classifications: o Measures of Absolute Dispersion - Expressed in the original unitsof the original observations
- Difference between the largestand the smallest values - Maximum value minus minimumvalue - For grouped data, the range isdefined as the difference between the upper class limit of the highest classand the lower class limit of the lowest class - Simplest/roughest measure
The Range R =59 – 10 = 49 (for discrete) R= 59.5 – 9.5 = 50 (for continuous)
· It is a weak measure because ittells only the extreme values and does not provide information on the valuesbetween · It is greatly affected byoutliers · For open-ended frequencydistributions, the range cannot be computed
Ø For Population Ø For Sample
8, 6, 3, 4, 4 Then 5 5 Thus,
The Standard deviation is therefore: S = S = S =
Computational Formula for the Sample Standard Deviation forUngrouped Data
Thus,
Ø Approximating the StandardDeviation for Grouped Data where f x x = mean ofthe frequency distribution n = numberof cases
Themean of this FDT was computed to be x = 37.17. The standard deviation iscomputed as follows:
∑f Hence, Ø Computational Formula forApproximating the Standard Deviation for Grouped Data
· Since it is a function of themean, 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 dataset is added or subtracted by the same amount k, then the standard deviation ofthe new data set is the same as the standard deviation of the original dataset. · If each of the data set ismultiplied by a constant k, then the standard deviation of the new data set isequal to k times the standard deviation of the standard deviation of theoriginal data set.
· Coefficient of Variation (CV) - Ratio of the standard deviationto its mean - Especially useful when onecompares the variability of one data set with another data set having differentunits CV = s/ · Index of Qualitative Variation(IQV) - Dispersion measures forqualitative nominal or ordinal variable - If all the values f a variableare in one category, then there is no variation and IQV = 0 - If all the values aredistributed evenly across the categories, IQV = 1, maximum value IQV = k(N where k = numberof categories N = numberof observations
- Shows the degree of asymmetryor departure from symmetry of a distribution - Indicates also the direction ofskewness § Types of Skewness o Positively Skewed § Longer tail to the right § § This happens when MoMe
o Negatively Skewed § Longer tail to the left § § This happens when
Ø Pearson’s Coefficients ofSkewness where Mo =mode Me =median SD =standard deviation We prefer to useFormula 2 over Formula 1 in as much as the mode does not always exist. Me = 37.54 s =11.31 Thus, we say that thedistribution of scores is skewed to the left.
Ø Describes the relative flatness or peakness of a distribution · Platykurtic – relatively flat · · Mesokurtic – is between the platykurticand leptokurtic curves; approaches or look like the “normal curve” Ø Pearson’s Coefficient of Kurtosis where x x = thesample mean n = no.of cases SD = standarddeviation Observe that if: Ku < 3, then the distribution isplatykurtic Ku = 3, then the distribution is mesokurtic Ku > 3, then the distribution is leptokurtic compilation of Dr. Libot ## Related Articles
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