Descriptive Measures February 24, 2021 postadmin Post in Uncategorized MODULE 4MATHEMATICS AS A TOOL FOR BUSINESS AND FINANCE Lesson No.2Lesson TitleDescriptive MeasuresSpecific LearningOutcomesDuring the learning engagements, students should be able to:1. Relate basic concepts of averages and deviations to normaldistribution; and2. Compute for means and deviations, and areas under the normaldistribution function PROCESSINGA powerful tool in statistics is the normal distribution. It allows us to make inferences about thewhole population based on some characteristics of a sample.If we gather all the weights of 1,000 freshmen students in a given school, the values will tend tocrowd around the value that is representative of the weight of majority of the students. Thesituation is shown by the hypothetical histogram below.Mean, Median, andMode are closely locatedhere with each other…Symmetry Line/Median (Midpoint)Bell-like shape curveAdding more and smaller weight ranges to the histogram makes the highest points of the barsapproximate a curve. This is the normal distribution curve, which is more commonly known asthe bell curve.Check this out: Symmetry Line/ Median (Midpoint)Mean or average is located around hereHigher frequency (mode) seen right of (orgreater than) the median…Scores of around 1000 students in a 140-item testa. Is there a possibility that the curve is “not normal”? or the shape is not bell-likeshape? –yesb. Is there a possibility that the curve is not symmetric? –yesc. If there is a possibility that the curve is not symmetric, then provide situations orexamples that support your idea: Mean, Median, and Mode (Central Tendency measures) are not likelysimilar or close to one another.Mostly of the scores are either above average or below average (dependswhere the skewness is directed-if positively skewed or negatively skewed)Mode is higher/greater than the mean; or, Mean is higher/greater than themode; Median is lower/lesser than the mean and mode; etc. d. What is the role of the measures of central tendency to the curve?A measure of central tendency is a single value that attempts to describe a set of data by identifying the centralposition within that set of data. As such, measures of central tendency are sometimes called measures of centrallocation. They are also classed as summary statistics. The mean (often called the average) is most likely the measureof central tendency that you are most familiar with, but there are others, such as the median and the mode.The mean, median and mode are all valid measures of central tendency, but under different conditions, somemeasures of central tendency become more appropriate to use than others. In the following sections, we will look atthe mean, mode and median, and learn how to calculate them and under what conditions they are most appropriateto be used.Intermission trivia: When to use Mean, Median, and Mode…?There are times that you cannot use the measures of central tendency in some types (or levels) ofdata or variable and there are only best measures of central tendency that must be used inparticular data/variable: Nominal data (lowest level of measurement) is defined as data that is used for namingor labeling variables, without any quantitative value. It is sometimes called “named”data – a meaning coined from the word nominal.oExamples are: Gender, Sexes, Skin and Hair Color, Nationality, color of the eyes, etc.Since these data are just counted but with no any quantitative value, MODE is the best centraltendency measure… (How many boys and girls are in this class? How many students arehaving blue eyes? How many Paulinian students are naturally born as Filipinos? Etc) Ordinal data (2nd lower level of measurement) is a categorical, statistical data typewhere the variables have natural, ordered categories and the distances between the categories is not known. These data exist on an ordinal scale, one of four levels ofmeasurement.. (in short, they are categorized or ranked…ordered, hence its name“ordinal)o Examples are: Honor students, Arranging of my Grade 10 subject grades fromlowest to highest, order of siblings in the family, etc.Since these data are categorized and arranged orderly or accordingly, MEDIAN is the bestcentral tendency measure… The middle value or mid-point in the ranked data gives us theCENTER point value of the group of data… Interval data (higher level of measurement) is a type of data which is measured along ascale, in which each point is placed at an equal distance (interval) from one another.Interval data is one of the two types of discrete data (Interval and Ratio). An example ofinterval data is the data collected on a thermometer—its gradation or markings areequidistant. Ratio Data (highest level of measurement) is defined as quantitative data, having thesame properties as interval data, with an equal and definitive ratio between each dataand absolute “zero” being treated as a point of origin. In other words, there can be nonegative numerical value in ratio data.Hence, the difference between Interval and Ratio data is that only Ratio accepts theABSOLUTE VALUE OF ZERO~ “Zero literally means nothing for Ratio…” but forInterval, Zero still has a value (example is the temperature… Zero degrees Celsius do notmean that there is absolutely no temperature but it means the freezing point of matter)Since these data are exact and quantitatively expressed, MEAN and MEDIAN are the bestcentral tendency measures…In Statistical distribution, you cannot use Median for skewed distribution because the scoresare not concentrated in the middle point (see previous examples in the processingportion)…Median is only best if the curve is NORMAL (bell-shape) since the concentratedscores are in the middle point.THE z-scoreThe z-score measures how many standard deviations an observed score is from the mean. Tocompute for the z-score, we use:̅̅Please use the URL to watch videos on solving z-score, or simply scan the QR codes using yoursmartphones to follow the link:ck12.org normal distribution problems: z-scoreBy Khan Academyhttps://www.youtube.com/watch?v=Wp2nVIzBsE8Activity 1: Computing for ZConsider ̅ and , get the z… 3834302624 a. What is the meaning of z in relationship with the mean and standard deviation? And viceversa?b. What happens when any value x is equal to the mean?c. What happens when z is equal to zero?Activity 2: Z and Normal CurveThe following table translates z-scores to their corresponding areas under the normal curve: z-scoreAreaz-scoreArea0.00.00001.60.44520.20.07931.80.46410.40.15542.00.47720.60.22572.20.48610.80.28812.40.49181.00.34132.60.49531.20.38492.80.49741.40.41923.00.4987 a. What is the role of z-score in relationship to the area under the normal curve?b. How can you define the area under the normal curve?c. What happens to the area when the z-score is negative?Search online for a z-table to compute for the area of following z-scores: z-scoreArea-1.642.891.750.010.49-0.330.19-2.55-1.011.11 Please use the URL to watch videos on how to use z-table or normal distribution table, or simplyscan the QR codes using your smartphones to follow the link:Normal Distribution Table – Z-table IntroductionBy Jalayer Academyhttps://www.youtube.com/watch?v=lgwT6tDniko&t=4sLearn how to create a normal distribution curve given meanand standard deviationBy Brian McLoganhttps://www.youtube.com/watch?v=qEGYkkif6xUAssessment1. The average number of workbook copies sold at a stand is 80 and a standard deviation of4. If the number of workbooks sold over a month follows a normal distribution,determine the probability that 25 copies are sold2. The following are the weights (in kgs) of students in a Math class of 40 students: 43453543384244374042383545393836384541374042354138374145404441444035413545404039 Construct a normal distribution curve and then solve for:a. Mean and Standard Deviationb. What is the probability that the person to be called has a weight of 42 kgs?SourcesMellosantos, Luiz Allan, et at. (2016). Math Connections in the Digital Age: Statistics andProbability. Sibs Publishing House, Inc. Quezon City.YouTube Presentation and Tutorialsck12.org normal distribution problems: z-score by Khan AcademyDescriptive Statistics in Excel by Elliott Jardin Ph.D.Excel Formulas: NORM.DIST (NORMDIST) by Brian HenryExcel Random Number Generator by Barb HendersonHow to Calculate Mean and Standard Deviation in Excel? by Eugene O’LoughlinLearn how to create a normal distribution curve given mean and standard deviation byBrian McLoganNormal Distribution – Explained Simply (part 1) by how2statsNormal Distribution Table – Z-table Introduction by Jalayer AcademyProperties of a Normal Distribution by Steve MaysStandard Deviation – Explained and Visualized by Jeremy JonesStatistics Fundamentals: The Mean, Variance and Standard Deviation by StatQuest withJosh StarmerStatQuest: The Normal Distribution, Clearly Explained!!! by StatQuest with Josh StarmerUsing NORM.DIST by Jeff DavisDeveloped by: JOENAMER P. OÑEZ (ACLC College of Butuan)MARY ANN M. GOZON (Butuan Doctors’ College)