Statistics and probability

Answer the following activities.

Set B: The following frequency distribution table shows the scores of 40 students in their 107-item Final Examination in Guidance and Counseling. Class Class Frequency Class Deviation Interval Boundaries (0 Frequency Mark (mp) fmp (d) Measures of Centrali Problem Set Points S stem LONG EXAM #1 Activit 8 Points System 3 point for each correct column (0 point for a single mistake in the column) Deviation Mean 10 points for each correct Median measures of centrality with correct computation Provisions for Mean, Median, and Mode Indicators 1 point for each correct measures of centrality without/wrong computation; 0.5 point for each measure of centrality with computation but wrong answer; and 0 oint for no answer and no com utation. Page 10 of11 Concept: LESSON 1: DESCRIPTIVE STATISTICS Descriptive statistics involves tabulating, depicting, and describing the collected data. The data are summarized to reveal overall data patterns and make them manageable. In a research study, we may have several measures or we may measure a large number of people on any measure. Descriptive statistics helps us to simplify large amounts of data in a sensible way. The Unlvariate Analysis of Data The univariate analysis involves the examination across cases of one variable at a time. There are three major characteristics of a single variable that we tend to look at: (a) the distribution, (b) the measures of central tendency, and (c) the measures of dispersion (Subong, Jr., 2005). In this module, we will only discuss the rst two major characteristics. LESSON 2: DISTRIBUTION The distribution is a summary of the frequency of individual values or ranges of values for a variable. The distribution of data has something to do with the organization of data which is a requirement in the processing of data so that it can be easier to analyze, interpret, to conclude. and to recommend the desired outcome to formulate a decision. One of the most common ways to describe a single variable is with a frequency distribution. To construct a frequency distribution, you should rst identify the lowest and highest value in the list. The difference between the highest and the lowest value is called the range. Page 1 of11 Activity 4: Arithmetic Mean Directions: The following are test scores in Science of the two sections from a 50-item test. Calculate the arithmetic mean and find out what section performs best. You may use a calculator for this activity. Write legibly and avoid erasures. Section A: 34, 42, 32, 38, 37, 31, 40, 29, 30, 45, 36, 32, 28, 37, 41 What section Section B: 40, 31, 29, 28, 41, 37, 30, 45, 42, 34, 32, 38, 37, 32, 36, performs best? Mean The mean may be computed by the midpoint method if the scores are many, say more than 30, and are grouped into a class frequency distribution. Use the formula: M = Efmp or - 2fmp N N Where: M or X = the mean = summation = frequency mp = or x', the midpoint Efmp = or Efx', the total products of the class midpoints and their frequencies N = the number of cases or scores (or population) Example: From the frequency distribution table, we need to look for the frequency and midpoint to compute for the fmp or the products of the class midpoints and their corresponding frequencies by multiplying the frequency and the class mark (or midpoint). For example, the frequency of the class interval 45-49 is 2 and its midpoint is 47, multiply 2 and 47 we have 94. Then get the "summation of fmp" (Efmp) by adding all the products in the fmp column, we have 1450. In this case, we have 50 cases or scores, N = 50. Class Class Interval Frequency (f) Class Mark Boundaries fmp (mp) 45 - 49 44.5 - 49.5 2 47 94 40 - 44 39.5 - 44.5 0 42 0 35 - 39 34.5 - 39.5 12 37 444 30 - 34 29.5 - 34.5 13 32 416 25 - 29 24.5 - 29.5 10 27 270 20 - 24 19.5 - 24.5 5 22 110 15 - 19 14.5 - 19.5 AC 17 68 10 - 14 9.5 - 14.5 4 12 48 N = 50 Efmp = 1450 Then simplify using the formula for the mean, (Efmp / N), 1450 divided by 50 is equal to 29. The mean 29 is the mean of the frequency distribution. If the mean is the yardstick (basis) of determining passing scores, 29 is passing. Based on the Mean = 29, 28 (56%) students passed the test, 22 (44%) students failed (based on the raw scores). Activity 5: Mean Directions: Find the weekly mean wage of 20 employees: Wage No. of Employees Computation: Php 1000-2000 3 Php 2000-3000 10 Php 3000-4000 NOT Php 4000-5000 Median The median is the point in a scale which divides the scale into two equally. A scale is a succession of numbers, steps, classes, degrees, gradations or categories with a fixed interval. The median computed from ungrouped scores is called crude, rough or counting median; while from a class frequency distribution, it is called a refined median. Page 6 of 11Activity 3: Frequency Distribution Table (Quiz 2, 30 points) Directions: Analyze the given situation and make a tabular representation out of it. Your research table should have a class size of 5, and a column for the class interval, frequency, percentage, midpoint, and cumulative frequency. You may refer to the sample table format and style indicated in the previous sample research table. Write legibly and avoid erasures. Be guided by the rubric below for scoring your work. In a survey of 20 patients who smoked, the following data were obtained. Each value represents the number of cigarettes the patient smoked per day. 10 8 6 14 13 11 18 14 13 12 15 17 15 11 16 Table Title: Quiz 2 Rubric (Activity 3, 30 points) Criteria Description HPS Score Sufficiency of the The output satisfies what is asked from the tasks Content 10 (research table) Appropriateness of |The output contains appropriate content and 10 the Content correlates with the research data. Mechanics and The document is organized; free of errors, 10 Organization grammar, spelling, and punctuation. Total 30 130 LESSON 3: MEASURES OF CENTRAL TENDENCY How do you analyze now the data on the frequency distribution? The central tendency has been defined by the authors as the tendency of the same observations or cases to cluster about a point, with respect either to an absolute value or to a frequency of occurrence; usually but not necessarily, about midway between the extreme high and extremely low values in the distribution. In simpler terms, a measure of central tendency is either a midpoint, an average, or the most frequent score in a distribution of scores. The most common measures of central tendency are the median, the mean, and the mode. Arithmetic Mean The arithmetic mean, or simply mean, is the average of a group of scores. The mean is easily affected by the magnitudes of the score. For instance, the mean of scores 1, 2, 3, 4, and 5 is 3. If 5 becomes 6, the mean becomes 3.2. Another example, the following are test scores in Biology: 25, 38, 41, 68, 71, 52, 64, 30, 45, 35, 58. Use the Formula: M = EXIN Where: M = the mean EX = the total of the scores N = the number of cases or scores (or population) In this case, just add all the scores divided by the number of cases (students), the total is 527 divided by 11, thus the mean is 47.91. Page 5 of 11Example of crude median: 45, 43, 40, 37, 33, 31, 28, the median is 37; the middle number. How if the set of scores is even, for example, 45, 43, 40, 37, 35, 33, 31, 28? Get the two middle numbers, add then divide it by 2; 37 plus 35 equals 72, 72 divided by 2 equals 36. For the refined median, use the formula: Man = LL + ( 2 - cf ) : Where: Man = the median LL = or EL, the exact lower limit N = the number of cases or scores (or population) cf = or cfb, the cumulative frequency equal to or next lower than N/2 = or fm, the frequency of the median class = the interval Example: From the frequency distribution table, we need to look at the location of the 0 deviation, from the class interval to the cumulative frequency. Cumulative Class Class Frequency Class Mark Deviation Frequency Interval Boundaries (f) (mp) (cf) by adding successively the frequencies from the highest to the lowest interval. In this case, we will just use the