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coming from a background understanding of the concept of function, shall be made aware that aformula exists for a function whose graph is the normal curve.f ( x) 12 2e( x )2 2 2In this formula, e is a mathematical constant whose approximate value is 2.1783, and π is aconstant whose approximate value is 3.1416. The variable x represents a score, μ stands for themean of the set of data, σ for standard deviation, and σ2 for the variance of the distribution. Themean μ and standard deviation σ are constant values obtained using the whole set of populationscores on their respective formulas; they are called parameters of the population.The function formula shown above is not commonly used to evaluate the normal densityfunction; the process involves laborious work even for students in postsecondary study. Instead,statistical tables giving the areas under the standard normal curve for standard scores that can beconverted into or from any observed scores from a set whose mean and standard deviation can beeasily determined by formula. These tables are regular appendices in statistics handbooks.Calculators are also programmed to give values for mean, standard deviation, or area.To underscore an earlier description, a normal distribution of scores assumes the shape of abell. A bell is symmetric, peaks in the middle, and trails down towards the tail-ends on both sidesof the middle. A short trail down from the middle peak on either side, the bell changes the slopeof its curve. These points in opposite sides of the peak are called the inflection points. Brasedescribes the inflection points as the transition points at which the bell curve changes betweendownward and upward cupping (293). The tail-ends trail infinitely away from the middle, nevertouching the x-axis. Mathematically, the behavior of such a graph is called asymptotic and itindicates, at least conceptually, that there will endlessly be extremely lower and lower, andextremely higher and higher scores with negligible but nonzero frequency.In a normal distribution, a big proportion of the scores cluster around the middle value, calledthe median in the ascending order of scores. The middle value in the array of horizontal scores isalso called the mode of the distribution, because it is the x-coordinate of the highest point in thegraph. Using the normal density formula, the mode is that value of x or that score that will givethe maximum normal density function value, which in turn corresponds to the highest frequencyin the distribution. In everyday intuitive language, when people talk of the average as somethingthat is commonly existing, they are most probably referring to the mode. The middle value isalso called the mean, which is the mathematical average of the scores obtained by dividing thesum of the scores observed by the total number of score data. This arithmetic mean is easy toinfer from the symmetry of the curve, which has identical densities on both sides of the verticalline passing through the peak and middle point. Again, if the distribution is normal, then mostscores are proximate to the average on both sides and then would symmetrically show descendingfrequencies as they assume values farther from the average. Figure 2 below summarizes theattributes and parameters of the normal curve.56Houston Teachers Institute

Bell or normal curves maydiffer from each other by their“peakedness” or flatness, that is,some normal curves may haveflatter peaks than others. Themathematical term for such acharacteristic of the normal curveis kurtosis, and a formuladetermines its value. The kurtosisof a normal curve is related to thespread of scores, or the deviationsof scores from the mean. Two setsFigure 2of scores on the same attributemay have the same middle value or mean, but given different standard deviations, the one withthe bigger standard deviation may show a greater spread of scores with a consequent flatter curvearound the mean, while the other will indicate a more peaked curve implying a higher frequencyof scores near the mean but a narrower spread of scores.In Figure 3, two hypothetical setsof raw scores in a test taken by twoclasses is represented by overlappingnormal curves showing the same meanof 80, but different spreads of scores,5 and 10, for the two classes.Regardless of peakedness andflatness, or with any kurtosis, the totalarea under any normal curve is alwaysequal to one. This is because the totalarea under it represents theproportions of scores added togetherfor an observed entire population, orFigure 3for a reasonably large sample that canbe accepted to represent and assumethe normal distribution character of the population from which it is taken. A shaded segment ofarea under the curve represents the proportion of scores falling within the specified range of scorevalues. Another interpretation is that the shaded area represents the proportion of the populationor sample possessing the includedrange of scores in this shaded area.Alternatively, the total area underthe curve represents the total ofprobabilities for the scores and isalways equal to one. Hence, an areasegment under the normal curve isthe probability that a specific rangeof scores will be manifested by anymember of the observed populationpicked at random. Figure 4 relatesarea to the density of scores or toFigure 4the probability of a specificobservation (score).Editha E. Josue57

As a model for the distribution of observed data, the normal curve depicts the empirical ruleof distribution. It is an empirical rule because it reflects the distribution from observed practice.This empirical rule has the following distribution of scores:1. Sixty-eight per cent, 68 %, of the scores are one standard deviation, or one scale unit,above or below the middle value or the mean.2. 95.44% of the scores are within two scale units on both sides of the middle value.3. 99.7% of the score data, practically including the whole population measured, fall withinthree standard deviations from the mean.Figure 5The scale along the horizontal axis in the normal distribution figure in Figure 5 clearly showsthe relationship between the mean μ and the standard deviation σ. Students in the course shallhave earlier gotten acquainted with the concept and computation of the mean of a data set.Knowing the mean, students get to understand the concept of variance. Computationally, thevariance is the average of the squared deviations of the scores from the mean of scores. Thesquare root of the variance will give the standard deviation of the distribution, the length of eachof the equal intervals dividing the scores. As a statistic, the standard deviation gives informationon the spread of scores of examinees. It will show how disparate or proximate these scores are.As mentioned earlier, a highly disparate set of scores would give a bell curve that has a wider butlower peak in the middle, and stretches out more widely to both tail-ends. On the other hand,using the same coordinate plane, the bell curve of a more proximate set of scores having the samemean, would have a steep peak in the middle and scale down more abruptly toward the tail-ends.Below is a summary of the formulas used in the process of constructing the bell curve for apopulation or sample of test-takers.For a set of n individual scores or observations where x 1, x2, x3 .xn, are the respectiveindividual scores or observations, the arithmetic mean or average is obtained using the formula:Mean:58μ or x ΣxnHouston Teachers Institute

If some or all of the individual observation measurements or counts exist more thanonce, then respective frequencies are taken into account in the formula for the mean:kμ or x kΣ 1 fi x i Σ 1 fi x inΣfIn this formula, n represents the total number of scores or observations, i refers to a specificscore with a specific frequency f, and k represents the number of different values of x in the set ofdata; k also refers to the number of products fixi.The notation Σ means sum and in the above formula, it denotes the sum of the products ofscores and their frequencies for the numerator part of the right side expression as well as the sumof the frequencies of the different scores.While the mean is the arithmetic average that measures the central tendency of a normallydistributed set of scores, the variance is the arithmetic average of the squared deviations of theindividual scores from the mean.Varianceσ2 Σ (x - x)2nAgain, the standard deviation determines the scale parameter for the normal curvedistribution of the set of scores or observations. Standard deviation, denoted by σ, is obtained bycomputing for the square root of the variance.σ Σ (x - x)2nThe above formula gives the population standard deviation. This is used when computingfor the standard deviation of the scores of all test-takers in a specific subject and level of theTAKS. The formula may also be used for a sample of 30 or greater number of observations fromthe population. For a sample size less than 30 (that is, if number of observations is less than 30),the formula above tends to underestimate the actual standard deviation of the population. Tocorrect for this negative bias, the sample standard deviation instead uses the more approximatevalue from the formula:σ Σ (x - x)2n-1Any normal curve may be converted into a standardized scale so that scores on two differenttests with differing scoring systems but relating to the same attribute may be compared. We mayconvert the differently scaled scores measuring the same attribute of the same population or groupinto the standard z-score using the formula below:Standard scorez x-μσwhere X is the scaled score based on a test‟s scoring system, μ is the arithmetic mean of scoresbased on that scoring system, and σ is the standard deviation of scores in that scoring system.Editha E. Josue59

In a standard normal curve, the mean is assigned the conversion value of 0, and the standarddeviation or scale interval is 1. Scores to the left of the zero mean will have negative values andscores to the right will have positive values.The Normal Curve of Standardized TestsThe Texas Assessment of Knowledge and Skills (TAKS) is a criterion-referenced test thatseeks to measure proficiencies in the areas of reading/language arts, mathematics, science, andsocial studies at Grades 3-11. Eleventh graders take the exit level TAKS the passing of which ispart of the requirement for graduation from high school. The percentage of students reaching orsurpassing the proficiency scale mark in these tests across academic areas and across grade levelsprovides the basis for a school‟s meeting or failing to meet the adequate yearly progress (AYP) asdefined in the NCLB accountability. Starting from a statistically determined baseline, thesepercentages of passing scores are evaluated against recommended yearly incremental increases(“Texas Consolidated State Application Accountability Workbook” 10-11).TAKS scores fall under three categories or levels: Did not Meet Standards, Met Standards,and Commended Performance. These categories correspond to the following levels ofproficiency: basic, proficient, advanced. Just to give an idea about the TAKS score scale, resultsin the TAKS 11th grade mathematics taken by students in 2006 indicate the proficiency level at2211 score mark, equivalent to a maximum of 27 wrong answers, and commended performancelevel starts at the 2400 mark, equivalent to at most seven wrong answers out of the total of 60 testitems. While scores signify the failure on, achievement, or surpassing of the proficiency mark,the normal curve also becomes a graphic analytical tool of information on a student‟sperformance in comparison to the population of test-takers (“Texas Assessment of Knowledgeand Skills: Raw Score Conversion Table, Mathematics - Spring 2006 Administration, ExitLevel”).For special populations who are beneficiaries of inclusion programs in regular public schools,the State-Developed Alternative Assessment (SDAA) and the Locally Determined AlternateAssessment are alternative assessment tools that replace the TAKS when the latter is deemedinappropriate to measure their gains in proficiency. Similarly, for students with limited Englishproficiency, the Reading Proficiency Tests in English (RPTE) is the standardized test deemedappropriate for assessment.The HISD has also been a participant to the Stanford and Aprenda national standardized testssponsored by the National Assessment of Educational Progress (NAEP) since 1997. TheStanford Achievement Test assesses student achievement in reading, mathematics, language,spelling, study skills, thinking skills, listening, environment/science, and social science. TheAprenda is the Spanish language version of the Stanford and is administered to Hispanic studentswith limited English proficiency. The SAT/Aprenda is a series of 11 steps that track students‟academic achievement from primary years to eleventh grade. As such, score reports also includean indicator on a test-taker‟s grade level in skills and knowledge. The Stanford AchievementTests are also norm-referenced, so that, again, the normal curve plays a key role in assessing thestate of American education, as individuals, schools, and groups are compared against thenational norm (“Glossary: Definition of Terms and Data Sources”).The Scholastic Aptitude/Assessment Test, or SAT, gives raw scores on a scale ranging from200 to 800 with a mean of 500 and a standard deviation of 100. But scores are reported inrelation to a norm group, or a similar group of test-takers, and are, therefore, converted into astandard score for college admission purposes. The SAT consists of two sections: verbal andmathematical reasoning abilities, and student‟s knowledge in specific subject areas and thestudent‟s ability to apply the knowledge (Samuda 131-133).60Houston Teachers Institute

Standardized test results are oftentimes reported along with percentiles and percentile ranks,which give information on the performance of the taker in comparison to the population of testtakers, or in accordance with the scores of a predetermined norm group that has been establishedas representative of the group of test-takers. If a particular examinee‟s score is determined to bein the 75th percentile, his or her percentile rank of 75 puts him/her in a position above 75% of theexaminees. In a normal distribution, when a student‟s or test-taker‟s score has been converted to az or standard score, his or her percentile rank among the population of test-takers may beobtained by referring to a standard or unit normal table. Such a table is normally found in theappendix section of a statistics reference book. For a start, the reader may refer to Gravetter andWallnau‟s “Statistics for the Behavioral Sciences (690 – 693).By definition, percentile ranks can be obtained for any distribution, whether normal orskewed (not symmetrical), in a set of data arranged in ascending order. The special nature of thedistribution of a normal curve makes scale scores that are assumed to be normally distributedeasily convertible into the percentile rank associated with standard scores. First, an examinee‟sscaled score is converted into a standard z score. The area to the left of this standard score,representing the probability or proportion of scores below that score, is determined using thenormal table for areas, or using a probability calculator. This probability or area is thenmultiplied by 100 to obtain the percentile rank based on an assumption of normal distribution ofachievement scores.Pr 100(Area to the left of z score)Since the percentile rank for any examinee is determined by his or her position in the rankedorder of the population of examinees, it can be obtained from a frequency table that arrangesscores, with their corresponding frequencies, in ascending order. Percentile rank for a particularscale score is obtained by adding the proportion of scores below that score to half the frequencyof that score. The sum is a fraction or decimal, which is then multiplied by 100 to give thepercentile rank. TEA‟s Student Assessment Division illustrates this procedure for obtaining thepercentile rank of a student directly from the frequency table of scaled scores for the 2003 exitlevel Grade (“Converting a TAKS Scale Score to a Percentile Rank”). The formula is worked outindependently of the mean μ and standard deviation σ:PR(x) ((f / 2 L ) / N)100x scale score of interestf frequency of the scale score of interestL cumulative frequency associated with the next lowest scale scoreN population size (number of persons tested).In a normal distribution, scores are closer around the mean and much farther apart at theextreme ends. Percentile ranks derived from a standard normal table would not be scaled onequal intervals the way the above formula based on a frequency table would generate their values.Normal table percentile ranks would show values that are more proximate around the mean andmore disparate at extreme scores. The normal curve equivalent (NCE) corrects for thismisleading ranking. The ranking according to the normal curve is modified such that thecalibrated scores along the horizontal axis are rescaled into 99 equal divisions with 50 as themiddle value and 1 and 99 as the first and last percentile ranks. The Stanford/Aprenda tests ofachievement utilize the NCE when ranking the population of examinees, because scores areinterpreted in comparison to a preexisting norm representing a nationwide population.Editha E. Josue61

Figure 6: (“Normal Curve Equivalent.” p/rdgstudy/nce.html)Performances of students in achievement and proficiency tests are always disaggregated intothe performances of subgroups of the population of test-takers, evaluated against the same criteriaof target gains and percentage of those passing proficiency. In the accountability system of theNCLB Act, these subgroups are the ethnic subpopulations (whites, African-Americans, Hispanics,Asians, Native Americans), special education populations, those with limited English proficiency(LEP), and those who fall under the definition of low socio-economic status (SES). Under theTexas state accountability system, comparison of the performance of demographic and programgroups is done on TAKS scale score averages and the percentages of students meeting standards.SAT and ACT mean scores for demographic groups are compared to the norm or to each other toassess differences and deviations from each other. The Stanford/Aprenda is likewise normreferenced.There is great concern about closing of the achievement gap among these groups. This hasalways been the ultimate standard or indicator by which the success of educational reforms isgauged. Very recently, Dr. Abelardo Saavedra, HISD‟s Superintendent, reminded allstakeholders to students‟ education, that while gains have been made in terms of the percentage ofstudents performing satisfactorily in TAKS and Stanford/Aprenda, there continues anunacceptably large achievement gap among the ethnic and disadvantaged groups that need to bebridged completely (“Achievement Gap Among Fifth-Graders in HISD Narrows”).In all these ways of assessing, the normal distribution of achievement is an underlyingassumption. This means that demographic groups and general populations follow a distributionwhere majority of the students in a group perform around a norm or central average. If a normaldistribution always depicts a set of data, what does it mean then to close the achievement gap?The integration of the mathematical concept with the goals of accountability is the challengethat students in the magnet program course will address. While the tests norm groups andcompare group averages, individual scores indicate whether students are performing at par withstandards or are performing better. The implication of a scenario where a growing number ofstudents have been meeting standards or achieving higher scores in these tests would translateinto group score means, the peaks in subpopulation normal

THE BELL CURVE: A MATHEMATICAL MODEL FOR A VARIETY OF REAL-WORLD PROCESSES The Nature of the Normal Curve and the Normal Curve of Nature The normal curve is a ubiquitous model in the subject of intelligence and achievement measurement, and is, therefore, a fixture, too, in an introducto