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LOGO Application of SPSS Part -1: Descriptive Statistics LOGO Irony of statistics Two statisticians were travelling in an airplane from Karachi to Islamabad. About half an hour into the flight, the pilot announced that they had lost an engine, but don't worry, there are three left. However, instead of 2 hours it would take 4 hours to get to Islamabad. A little later, he announced that a second engine failed, and they still had two left, but it would take 5 hours to get to Islamabad. Somewhat later, the pilot again came on the intercom and announced that a third engine had died... Never fear, he announced, because the plane could fly on a single engine. However, it would now take 8 hours to get to Islamabad. At this point, one statistician turned to the other and said, "Gee, I hope we don't lose that last engine, or we'll be up here forever!" LOGO Contents 1 What is descriptive research? 2 Types of descriptive measures LOGO What is descriptive statistics? Descriptive statistics aim at describing a situation by summarizing information in a way that highlights the important numerical features of the data. A good summary captures the essential aspects of the data and the most relevant ones. It summarizes it with the help of numbers, usually organized into tables, but also with the help of charts and graphs that give a visual representation of the distributions. LOGO Types of univariate descriptive measures There are three important types of univariate descriptive measures: measures of central tendency, measures of dispersion, and measures of position LOGO measures of central tendency Sometimes called measures of the center. It answer the question: What are the categories or numerical values that represent the bulk of the data in the best way? Such measures will be useful for comparing various groups within a population, or seeing whether a variable has changed over time. Measures of central tendency include: the mean (which is the technical term for average), the median, and the mode. LOGO Measures of dispersion Measures of dispersion answer the question: How spread out is the data? Is it mostly concentrated around the center, or spread out over a large range of values? Measures of dispersion include: the standard deviation, the variance the coefficient of variation. LOGO Measures of position Measures of position answer the question: How is one individual entry positioned with respect to all the others? Or how does one individual score on a variable in comparison with the others? If you want to know whether you are part of the top 5% of a math class, you must use a measure of position. Measures of position include: percentiles, deciles, and quartiles. LOGO Should NRO be allowed? LOGO Standard Deviation LOGO Standard Deviation and Probability In general, people use the +/- 2 SD criteria for the limits of the acceptable range for a test When the measurement falls within that range, there is 95.5% confidence that the measurement is correct Only 4.5% of the time will a value fall outside of that range due to chance; more likely it will be due to error LOGO Example Consider the following three datasets: (1) 5, 25, 25, 25, 25, 25, 45 (2) 5, 15, 20, 25, 30, 35, 45 (3) 5, 5, 5, 25, 45, 45, 45 LOGO Solution Case Standard Deviation 1 11.55 2 13.23 3 20.00 The standard deviations for the datasets are 11.55, 13.23, and 20. The larger standard deviations indicate greater variability in the data, and in general we can say that smaller standard deviations indicate less variability in the data. LOGO Example 4 For example, each of the three populations {0, 0, 14, 14}, {0, 6, 8, 14} and {6, 6, 8, 8} has a mean of 7. LOGO Solution Their standard deviations are 7, 5, and 1, respectively. The third population has a much smaller standard deviation than the other two because its values are all close to 7. In a loose sense, the standard deviation tells us how far from the mean the data points tend to be. It will have the same units as the data points themselves. If, for instance, the data set {0, 6, 8, 14} represents the ages of a population of 4 cows, the standard deviation is 5 years. LOGO Example 5 Consider average temperatures for cities. While two cities may each have an average temperature of 15 °C, it's helpful to understand that the range for cities near the coast is smaller than for cities inland, which clarifies that, while the average is similar, the chance for variation is greater inland than near the coast. So, an average of 15 occurs for one city with highs of 25 °C and lows of 5 °C, and also occurs for another city with highs of 18 and lows of 12. The standard deviation allows us to recognize that the average for the city with the wider variation, and thus a higher standard deviation, will not offer as reliable a prediction of temperature as the city with the smaller variation and lower standard deviation. LOGO Standard Deviation For example, the average height for adult men in Pakistan is about 70 inches, with a standard deviation of around 3 in. How we would interpret it? LOGO Interpretation This means that most men (about 68 percent, assuming a normal distribution) have a height within 3 in of the mean (67–73 in) – one standard deviation, whereas almost all men (about 95%) have a height within 6 in (15 cm) of the mean (64–76 in) – 2 standard deviations. LOGO If the standard deviation were zero, then???? LOGO …then all men would be exactly 70 in high. LOGO If the standard deviation were 20 in, then …??? LOGO …men would have much more variable heights, with a typical range of about 50 to 90 in LOGO Sigma zσ percentage within percentage outside ratio outside 1σ 68.2689492% 31.7310508% 1 / 3.1514871 1.645σ 90% 10% 1 / 10 1.960σ 95% 5% 1 / 20 2σ 95.4499736% 4.5500264% 1 / 21.977894 2.576σ 99% 1% 1 / 100 3σ 99.7300204% 0.2699796% 1 / 370.398 3.2906σ 99.9% 0.1% 1 / 1000 4σ 99.993666% 0.006334% 1 / 15,788 5σ 99.9999426697% 0.0000573303% 1 / 1,744,278 6σ 99.9999998027% 0.0000001973% 1 / 506,800,000 7σ 99.999 999 999 7440% 0.0000000002560% 1 / 390,600,000,000 LOGO LOGO Standard Error of Mean. A measure of how much the value of the mean may vary from sample to sample taken from the same distribution. It can be used to roughly compare the observed mean to a hypothesized value (that is, you can conclude the two values are different if the ratio of the difference to the standard error is less than -2 or greater than +2). 25 LOGO Coefficient of Variation LOGO Coefficient of Variation The Coefficient of Variation (CV) is the standard Deviation (SD) expressed as a percentage of the mean Also known as Relative Standard deviation (RSD) CV % = (SD ÷ mean) x 100 LOGO month-wise average temp (mm) Month Karachi Peshawar January 30 -1 February 31 4 March 32 25 April 33 35 May 34 40 June 35 48 July 35 50 August 34 45 September 33 38 October 32 35 November 31 25 December 30 4 Calculate CoV and see whether meaningful conclusion can be drawn 28 LOGO How to diagnose issues related with normal distribution LOGO Kurtosis Kurtosis value tells whether distribution is peaked, flat, or normal. If Kurtosis value is zero, distribution is normal, if it is positive, then distribution is more peaked than normal and if it is negative, then distribution is flatter than normal. Kurtosis values ranging from -1 to +1 are considered excellent. (George & Mallery, 2006, p. 98) 30 LOGO For a normal distribution, the value of the kurtosis statistic is zero Bell-shaped curves = describe in terms of its kurtosis (curvature) 1. Leptokurtic = thin distribution (concentrated at midpoint) (-) 2. Mesokurtic = normal distribution 3. Platykurtic = flat distribution (+) 31 LOGO The large positive kurtosis tells you that the distribution of data is more peaked and has heavier tails than the normal distribution. 32 LOGO Skewness Skewness value tells whether distribution is symmetrical or asymmetrical. If Skewness value is zero, distribution is symmetrical, if it is positive, then smaller values are in greater number in distribution and if it is negative, then larger values are greater in number in distribution. Skewness values ranging from -2 to +2 are acceptable. 33 LOGO Non-symmetrical 1. Positive Skew = high number of low scores 2. Negative Skew = high number of high scores 34 LOGO Skewness value = 0 35 LOGO Large positive skewness shows that sale has a long right tail. That is, the distribution is asymmetric, with some distant values in a positive direction from the center of the distribution. 36 LOGO Add your company slogan

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