MATHEMATICS APPLIED TO BIOLOGICAL SCIENCES MVE PA 07. LP07 DESCRIPTIVE STATISTICS - Calculating of statistical indicators (1)

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1 LP07 DESCRIPTIVE STATISTICS - Calculating of statistical indicators (1) Descriptive statistics are ways of summarizing large sets of quantitative (numerical) information. The best way to reduce a set of data and still retain its information is to summarize it with a single value. These values are known with the name synthetic indicators. With synthetic indicators are evaluate: 1. Central tendency 2. Measures of spread 3. Measures of shape Measures of central tendency Central tendency refers to the idea that there is one number that best summarizes the entire set of measurements, a number that is in some way "central" to the set. Measures of central tendency mean, median, and mode can help you capture, with a single number, what is typical of a data set. The mean is the average value of all the data in the set. The median is the middle value in a data set that has been arranged in numerical order so that exactly half the data is above the median and half is below it. The mode is the value that occurs most frequently in the set. In a normal distribution, mean, median and mode are identical in value. When not to use the mean. The mean has one main disadvantage: it is particularly susceptible to the influence of outliers. Statistical outliers are data points that are far removed and numerically distant from the rest of the points. These are values that are unusual compared to the rest of the data set by being especially small or large in numerical value. When is the mean the best measure of central tendency? The mean is usually the best measure of central tendency to use when your data distribution is continuous and symmetrical, such as when your data is normally distributed. If a Histogram of a series of data has a shape often referred to as a "bell curve." this series of data has a normal distribution. However, it all depends on what you are trying to show from your data. When is the mode the best measure of central tendency? The mode is the least used of the measures of central tendency and can only be used when dealing with nominal data. For this reason, the mode will be the best measure of central tendency (as it is the only one appropriate to use) when dealing with nominal data. The mean and/or median are usually preferred when dealing with all other types of data, but this does not mean it is never used with these data types. When is the median the best measure of central tendency? The median is usually preferred to other measures of central tendency when your data set is skewed (i.e., forms a skewed distribution) or you are dealing with ordinal data. However, the mode can also be appropriate in these situations, but is not as commonly used as the median. Measures of spread or dispersion Dispersion refers to the idea that there is a second number which tells us how "spread out" all the measurements are from that central number. Measures of spread include range, quartiles, variance and standard deviation.

2 Range The range is very easy to calculate because it is simply the difference between the largest and the smallest observed values in a data set. Thus, range, including any outliers, is the actual spread of data. Range = difference between highest and lowest observed values Quartiles The median divides the data into two equal sets. The lower quartile is the value of the middle of the first set, where 25% of the values are smaller than Q1 and 75% are larger. This first quartile takes the notation Q1. The upper quartile is the value of the middle of the second set, where 75% of the values are smaller than Q3 and 25% are larger. This third quartile takes the notation Q3. Variance (S 2 ) = average squared deviation of values from mean The variance for a discrete variable made up of n observations is defined as: Standard deviation (S) = square root of the variance The standard deviation is the "average" degree to which scores deviate from the mean. The standard deviation for a discrete variable made up of n observations is the positive square root of the variance and is defined as: Measures of shape The histogram can give you a general idea of the shape, but two numerical measures of shape give a more precise evaluation: skewness tells you the amount and direction of skew (departure from horizontal symmetry), As we will see, data can be skewed either to the right or to the left. Kurtosis tells you how tall and sharp the central peak is, relative to a standard bell curve. Kurtosis is the measure of the peak of a distribution, and indicates how high the distribution is around the mean. Objectives: 1. Calculating the synthetic indicators using formulas 2. Interpretation of indicators of centrality spread and shape 3. Calculating the synthetic indicators as a result of a call to the statistical functions in Excel 4. Calculating the synthetic indicators as a result of a call to Descriptive statistics option of Data Analyses

3 Problema 1 The table below shows the concentrations of cholesterol in the blood, as measured in mg / dl for the sample of 50 patients Calculating the mean, median and modal value with: a. Formulas, b. Excel functions c. Option Descriptive statistics of Data Analyses - Calculating the range, variance, standard deviation, coefficient of variation with: a. Formulas, b. Excel functions c. Option Descriptive statistics of Data Analyses - Calculating skewness and kurtosis - Determine the quartiles Q1, Q2 and Q3 which divide the dataset into four sections with equal number of values. - Plotting box plot diagram Ind: To calculate synthetic indicators based on calculation formulas or using the functions implemented in EXCEL it is necessary to enter the values observed on a column in the spreadsheet a. Calculation based on formulas or definition: - For calculating the average, will sum with SUM function or AotoSum 50 values. - Enter the calculation formula for dividing the sum by If the series of observations is given in the form of frequency distribution, will take into account the frequency of occurrence of each value for average value calculation (m = x in i n i ) - The median value (Me) of a statistical series that divides the ordered sequence of variable values in two parts, each part containing the same number of values. To find this value proceed as defined: - will write to the values of one column (in Excel) - will sort these values in ascending order - if are many values (50 in our case), even number, 25th and 26th values are 200 or 200, so in this case Me = 200 (average of the 25th to 26th respectively in the string value of observations). - If the number of variables were odd, for example 51, the median value would have been the 26th of observations ordered in ascending value - To calculate the modal value is necessary to obtain the frequency distribution, the modal value is the value of variable that has the highest frequency of occurrence. - To calculate the amplitude is finding the maximum and minimum value required. They can be ordered ascending series of values. 2 ( xi m) 2 s - For the calculation of variance using the formula, N it is necessary calculation of m (arithmetic mean), then you use the Fill Down and Auto Sum.

4 - Standard deviation as the square root of the variance caculează - The coefficient of variance is calculated as: CV [%] = standard deviation aritmetic mean 100 b. To calculate requirements implemented using Excel functions will proceed as follows: - The column will be written as labels, the requirements problem. - In their own right, select the next cell and call functions that calculate requirements: AVERAGE, MEDIAN, MODE, VAR, STDEV, QUARTILE, SKEW, KURT, and so on, - All these functions have the argument the spreadsheet that contains the table of observations (table primary evidence or actual table) average =AVERAGE(A1:A50) 198,7 median =MEDIAN(A1:A50) 200 mode =MODE(A1:A50) 180 variance =VAR(A1:A50) 1319, Standard deviation =STDEV(A1:A50) 36, coefficient of variation =(B67/B63) % 0, skewness =SKEW(A1:A50) -1, kurtosis =KURT(A1:A50) 0, Q1 =QUARTILE(A1:A50,1) 170 Q2 =QUARTILE(A1:A50,2) 200 Q3 =QUARTILE(A1:A50,3) 230 c. All indicators can be calculated by simply calling the function Descriptive Statistics of Data Analyses (Data menu). Installing the Data Analysis option has been presented in previous work. From the dialog that appears, select Descriptive Statistics. Click on OK. You will see the results 1. In the input section, enter the data range in Input Range. (B2:B13 here.) 2. In the output section, enter the cells where you want to save the result. [Here the results are put in the same worksheet.] 3. Select Summary Statistics. 4. Then click on OK Mean 198,7 Standard Error 5, Median 200 Mode 180 Standard Deviation 36,32071 Sample Variance 1319,194

5 Kurtosis -1,16536 Skewness 0, Range 120 Minimum 140 Maximum 260 Sum 9935 Count 50 Confidence Level(95,0%) 10, Mean - arithmetic average. Can be calculated and the AVERAGE function. If the variable is normally distributed, the average interval indicates the middle of the minimum and maximum (range distribution data). Also in case of normal distribution around the mean (ie average within-standard deviation, mean + standard deviation are the most data. - Standard Error - Standard error. The standard error is involved in the calculation of 95% confidence interval about the mean ( only one variable regular distribution ), is also involved in the statistical inference. - Median - The median is the value of the series so that half of the observations have values less than ( or equal to ) and the other half have values greater than (or equal). It can calculate and MEDIAN function. If the normal distribution mean and median are equal. So the arithmetic mean median and indicators are normal distribution, how have values closer the more likely that the variable is normally distributed. The term " closest " is estimated by the size of the standard error. - Mode - The module is the value that has the highest frequency in the series. If a situation arises module is where the series is not the way, that all the values appear only once. When displayed value # N / A. Another possible situation is that the series is bimodal or tri-modal. Then only the first value will be displayed in order of their appearance in the series. In this case to determine all values module can make a frequency table. It can calculate and MODE function. The module is useful in the case of qualitative variables ordered, but in the case of other variables, for example, in the case of continuous variable with normal distribution module is likely to have a value apopiată medium. - Standard Deviation - The standard deviation can be calculated with the function STDEV and the population standard deviation with the function STDEVP. The standard deviation shows the mean square deviation is the arithmetic variable. If you have a small value, then the data varies slightly around the average. If the distribution is represented by Gauss parameters measuring central tendency (mean, median, modal value and the central value) have the same values. In this case (normal distribution) distributions occur following data: - Interval X 1 s contains about 68.3% of the observations - Interval X 2 s contains about 95.5% of the observations - Interval X 3 s contains about 99.7% of the observations - Sample Variance - Variation can be calculated with VAR or VARP function - The Skewness measure indicates the level of non-symmetry. If the distribution of the data are symmetric then skewness will be close to 0 (zero). The further from 0, the more skewed the data. A negative value indicates a skew to the left. How do you tell if the skewness is large enough to case concern. Excel doesn t give you this value, but a measure of the standard error of skewness can be calculated as =SQRT(6/N) or =SQRT(6/50) which is If the skewness is more than twice this amount, then it indicates that the distribution of the data is non-symmetric. In this case * 2 = The skewness reported by Excel is , so the data can be assumed to be fairly symmetric (although somewhat marginally so.) However, this does NOT indicate that the data are normally distributed. - Kurtosis is a measure of the peakedness of the data. Again, for normally distributed data the kurtosis is 0 (zero). As with skewness, if the value of kurtosis is too big or too small, there is concern about the normality of the distribution. In this case, a rough formula for the standard error for kurtosis is =SQRT(24/N) = Twice this amount is Since the value of kurtosis falls within two standard errors (-0.26) the data may be

6 considered to meet the criteria for normality by this measure. These measures of skewness and kurtosis are one method of examining the distribution of the data. However, they are not definitive in concluding normality. You should also examine a graph (histogram) of the data and consider performing other tests for normality such as the Shapiro-Wilk or Kolmogorov-Smirnov test (not provided by Excel). - Range - amplitude is maximum-minimum difference data series. - Minimum - The minimum value of the smallest in the series. It can calculate with MIN function - Maximum - The maximum amount of the highest in the series. It can calculate with MAX function - Sum - The sum or total values of the series. It can be calculated with SUM function. - Count - The number of observations or sample size n = 20. Can be calculated with COUNT function. - Quartiles and percentiles are similar median. The first quartile or value having the property that 25% of the data series are less than or equal to it, and 75% higher or equal to the first quartile. The second quartile is the median. The third quartile is a value having the property that 75% of the data series are less than or equal to it and 25% higher or equal to the third quartile. - Percentile in the order of a value is such that an amount of data is equal to or less while the other is larger. - CV=STDEVP/AVERAGE - Coefficient of variation: you can use the following rules of thumb for interpretation: - If CV is below 10% then the population can be considered homogeneous; - if CV is between 10% -20% if the population can be considered relatively homogeneous; - if CV is between 20% -30% if the population can be considered relatively heterogeneous; - if CV is above 30% then the population can be considered heterogeneous. For plotting box plot diagram recommend using the program online at: Problema 2. Is provided a study on a sample size of 20 patients. We collect data on the following parameters Biomedical: diastolic blood pressure (DBP) (mmhg), systolic blood pressure (SBP) (mmhg), age (days), height (cm), weight (grams), sex. Data are presented in Table Excel file SD. Save the file in your folder and make the following statistical processing this file. Requirements It aims to calculate - Indicators of central tendency (arithmetic mean, median, mode, geometric mean, harmonic mean, the central value) - Dispersion (amplitude, mean deviation, variance, standard deviation, coefficient of variation, standard error, asymmetry, flatness) and - Location (first quartile (minimum), second quartile, third quartile (median), the fourth quartile, the fifth quartile (maximum) for quantitative variables. - Insert table on Sheet1 Excel file Table SD. Calculate the required indicators in this table. To calculate statistical indicators used Excel functions or formulas - see instructions. Interpret the results statistically (based on the arithmetic mean, median, boltirii, asymmetry, standard error, standard deviation, minimum and maximum variables assess if the distribution is Gaussian normal distribution). Indicators of central tendency Denumire SBP DBP Age heigh t Aritmetic mean (AVERAGE) Median (MEDIAN) Mode (MODE) Geometric mean (GEOMEAN) wieg ht

7 Indicators of dispersion Indicators shape and location Armonic mean (HARMEAN) Range (RANGE) Standard deviation (STDEV) Variance (VAR) Coeficientul de variaţie (with formula!!!!) m-s INTERVALS - m represente arithmetic mean - s represent standar deviation Eroarea standard 1 m+s m-2s m+2s m-3s m+3s Skewness (SKEW) Kurtosis (KURT) First quartiel (min) (QUARTILE) Second quartile The third quartile (mediana) (QUARTILE) The fourth quartile (QUARTILE) A fifth quartile (maxim) (QUARTILE). 1. Calculate descriptive statistics (using Data Analysis) for the first 6 variables and interpret the results. 2. Format results: reduce the number of decimal places (two decimal) SBP (mmhg) DBP (mmhg) Age (zile) Height (cm) Weight (grame) Mean 143,68 76,05 47,42 54, ,32 Standard Error 5,20 3,78 4,17 0,74 165,94 Median Mode Standard Deviation 22,66 16,46 18,17 3,22 723,30 Sample Variance 513,45 271,05 330,15 10, ,89 Kurtosis -0,75-0,93-0,66-0,20 0,15 1 To calculate the standard error of the mean by itself, use the one of the following formulas = STANDARD DEVIATION / SQUARE ROOT OF THE POPULATION SIZE -or- = STDEV(range of values)/sqrt(number) where: The range of values is the data used to determine the standard deviation. -and- The number is the size of all the possible random samples.

8 Skewness -0,12 0,02 0,42 0,28-0,44 Range Minimum Maximum Sum Count Confidence Level(95,0%) 10,92 7,94 8,76 1,55 348,62 Problema 3 Age at death to a total of 60 dogs are shown below. 11,8; 3,6; 16,6; 13,5; 4,8; 8,3; 8,9; 9,1; 7,7; 2,3; 12,1; 6,1; 10,2; 8; 11,4; 6,8; 9,6; 19,5; 15,3; 12,3; 8,5; 15,9; 18,7; 11,7; 6,2; 11,2; 10,4; 7,2; 5,5; 14,5; 16,6; 13,5; 4,8; 8,3; 8,9; 9,1; 7,7; 2,3; 12,1; 6,1; 9,6; 19,5; 15,3; 12,3; 8,5; 16,6; 13,5; 4,8; 8,3; 8,9; 9,1; 7,7; 2,3; 12,1; 6,1; 9,6; 19,5; 15,3; 12,3; 8,5 - Calculate the mean, median and modal value. - Calculate the amplitude, variance, standard deviation, coefficient of variation. - Determine the quartiles Q1, Q2 and Q3 which divide the dataset into four sections with equal number of values. - Prepare frequency distribution, and perform histogram - Calculate the index of asymmetry and vaulting. Problema 5 The following data are the age (in years) of a virus infection has been diagnosed with a sample of 27 randomly selected cases: 39, 50, 26, 45, 71, 51, 33, 40, 40, 51, 66, 63, 55, 36, 57, 41, 61, 47, 44, 48, 59,42, 54, 47, 53, 54, Calculate accurate to two decimal places following statistics: - media - median - module - the amplitude - version - Standard deviation - Coefficient of variation 2. How many points are summarized in the following ranges: - X ± 1. s - X ± 2. s - X ± 3. s 3. Using your coefficient of variation, specify the uniformity of the series of observed values

9 Probleme propuse (opțional) 1. Consider the following series of frequency distribution: xi fa Calculations three parameters of central tendency; Determine the variance and standard deviation of the data set. What is the coefficient of variation? 2. A die was thrown 45 times. The frequency of numbers obtained are shown in the table below: Scor Frecvenţă What is the average score obtained? 3 The temperature was measured at ground level within 24 hours. The result is the following graph: What is the amplitude? a 10 F b 30 F c 40 F d 50 F 4. What is the amplitude for the next range of values? 57, -5, 11, 39, 56, 82, -2, 11, 64, 18, 37, 15, 68 a 11 b 68 c 77 d The table below shows the monthly average in 2013 Month Ian. Feb. Mar. Apr. Mai Iun. Iul. Aug. Sept. Oct. Noi. Dec. Temperatura medie ( C) Calculate the average annual temperature 2. Compute median 3. Compute the dominant value