none
Prepared by Rossi A. Hassad, PhD, MPH, CStat DESCRIPTIVE STATISTICS REPORT The objective of this analysis was to describe the distribution of the number of sex partners (in the past month) reported by a sample of college students. Data were collected from thirty students (N = 30). Measures of central tendency (mean, median, and mode), measures of dispersion (standard deviation, variance, and range) along with a bar chart and histogram were generated. The mean, median, and mode were 3, 1 and 0 sex partner(s) respectively (see table below). The difference among these values indicated a non-normal distribution, and at least 50% of the students reported 1 or more sex partner(s) 1 . This observation of non-normality was consistent with the standard deviation of 3 sex partners, which in this context is relatively high. Together, the histogram and bar chart revealed a multimodal distribution with a considerable degree of positive skewness. Also, it must be noted that the coefficient of skewness (see “skewness” in table below) is approximately 2, and generally, a coefficient greater than 1 indicates marked positive skewness. In conclusion (and from a practical perspective) this distribution consists of three subgroups; students who reported 0, 1 and multiple sex partner(s) in the past month. The positive skewness observed can be accounted for by the value 15, which in this context can be considered an outlier. It is recommended that this outlier (15) be removed and the data reanalyzed in order to obtain more reliable and representative descriptive statistics. 1 Also, at least 50% of the students reported 1 or no sex partner. Prepared by Rossi A. Hassad, PhD, MPH, CStat Descriptive Statistics for Number of Sex Partners Number of Sex Partners Reported by College Students N 30 Mean 2.67 Median 1.00 Mode 0 Std. Deviation 3.144 Variance 9.885 Skewness 2.120 Std. Error of Skewness .427 Range 15 Prepared by Rossi A. Hassad, PhD, MPH, CStat Bar Chart NOTE: Please submit your work as a single PDF file. Assignment #1 Describe the distribution of the number of hours of sleep below, obtained from a sample of 40 college students. Comment on the center, shape (including graphs), and spread, according to the format discussed in class (see attachments). DO NOT use the variable "Gender" for this assignment. NOTE: Manually (as well as using SPSS) calculate and present the appropriate measures of central tendency, measures of dispersion, and graphs. Write a brief report, using the attached format. You are required to submit the following: 1. The manual (by hand) calculations of the measures of central tendency and measures of dispersion. Use 2 decimal places for all calculations. 2. A manual (by hand) sketch of a graph (either a bar chart or histogram, with the axes labeled). 3. A brief (type-written) report. Please refer to the attached example. The report must include a table (with a title) and two graphs (a bar chart and a histogram, with the axes labeled) from the SPSS output. NOTE that you are NOT required to manually calculate the coefficient of skewness. For the report, use the skewness value from the SPSS output. Please upload your work as a single PDF file. Number of Hours of Sleep (per School night) Gender 4 Male 4 Male 5 Male 5 Female 5 Male 6 Male 6 Male 6 Male 6 Male 6 Male 6 Male 6 Male 7 Female 7 Female 7 Female 7 Male 7 Male 7 Male 7 Male 7 Male 8 Female 8 Male 8 Male 8 Female 8 Female 8 Female 8 Female 8 Female 8 Female 8 Female 8 Female 8 Female 9 Female 9 Male 9 Female 9 Female 9 Male 9 Female 9 Female 10 Female Rossi A. Hassad, PhD, MPH Do not use or distribute this material without prior written permission from Rossi A. Hassad, PhD, MPH (Rhassad@mercy.edu)- June2014 1 MODULE # 2 Statistics for the Social & Behavioral Sciences Performing and Interpreting Basic Descriptive Statistics A survey was conducted using a sample of 30 (thirty) students. Each student was asked to report the number of sex partners he/she had in the past month. The information collected is detailed below. 1 5 5 4 5 4 0 1 1 0 0 1 0 4 2 0 0 1 1 4 5 5 0 0 1 5 5 15 5 0 NOTE: In this exercise we will summarize the distribution of these data by calculating, interpreting, and reporting the statistical measures mentioned below. When we summarize, we provide the essentials. We will also represent the distribution using graphical formats (line graph, bar chart, and histogram). By doing this, we have both a numerical and a pictorial representation of the data, and together this allows for greater clarity and more effective communication of the results. The numerical measures alone do not tell the entire story of the data, therefore, do not forget the graph (or picture of the data). We all know how effective a picture can be in conveying information. Hence we will focus on the following: Measures of central tendency: Mean, Median, and Mode (CENTRE) Measures of Dispersion: Standard Deviation, Variance, and Range (SPREAD) Line Graph, Bar Chart, and Histogram (SHAPE) Recall from the notes that in order to adequately describe a distribution we must comment on its CENTER, SHAPE, and SPREAD. mailto:Rhassad@mercy.edu)- Do not use or distribute this material without prior written permission from Rossi A. Hassad, PhD, MPH (Rhassad@mercy.edu)- June2014 2 As you read these notes and other relevant material, you will observe that there are symbols that are used to represent various statistical measures. These differ for the sample (statistics) and the population (parameters). Sample measures are often assigned Roman letters (e.g. M, s, x̄ ), whereas the equivalent unknown values (parameters) in the population are represented by Greek letters (e.g. µ and σ) – see below. NOTE: The Greek letter upper case S) indicates summation (or total) in statistics. Measures Sample (statistic) Population (parameter) Mean M, x̄ µ (mu) Standard Deviation S, SD (sigma) mailto:Rhassad@mercy.edu)- Do not use or distribute this material without prior written permission from Rossi A. Hassad, PhD, MPH (Rhassad@mercy.edu)- June2014 3 MPORTANT: The steps in the calculations are detailed and explained below. PLEASE PERFORM ALL CALCULATIONS TO TWO DECIMAL PLACES. Column # 1 Column # 2 Column #3 Column # 4 Column # 5 Column # 6 These are the different values or observations (from the 30 listed above) – ranked from lowest to highest. Ordering the values is important especially for obtaining the MEDIAN (which is addressed below). f represents the frequency or how often each x value (from column 1) occurred. For example, 9 students reported 0 sex partner. We are grouping the data so that it is better and meaningfully organized. Therefore, we do not have to list each of the 30 values separately. This is the product of f and x, that is, multiplying each x value (from column 1) by the corresponding f value (from column 2). This helps us to get the sum of the 30 values in steps, which is more methodical and organized. This is the same as adding up the 30 values. x̄ (pronounced x bar) represents the mean of the sample , which as calculated below is 2.67 for these data. This column represents the deviation (or difference) of each x value (column1) from the mean (x̄ ). For example , 0 – 2.67 = -2.67 hence the first entry here is: -2.67 This is the square of each value in column 4. This is done so as to get rid of the negative signs and allow for meaningful summation of the values. Hence the first entry here is: -2.67 x -2.67 = 7.13 (recall that the square of a negative number is a positive value). In this column, each value from column # 5 is multiplied by its corresponding f value from column # 2. This is done so as to account for all the values in the distribution. Hence the first entry is 9 x 7.163 = 64.17 x f fx x - x̄ (x – 2.67) (x - x̄ ) 2 f(x - x̄ ) 2 0 9 0 -2.67 7.13 64.17 1 7 7 -1.67 2.79 19.53 2 1 2 -.67 .45 .45 4 4 16 1.33 1.77 7.08 5 8 40 2.33 5.43 43.44 15 1 15 12.33 152.03 152.03 f = 30 fx = 80 The arrow in the table above indicates the mode or modal value (which is explained below). mailto:Rhassad@mercy.edu)- Do not use or distribute this material without prior written permission from Rossi A. Hassad, PhD, MPH (Rhassad@mercy.edu)- June2014 4