DESCRIPTIVE STATISTICS WITH EXCEL

Transcription

1 DESCRIPTIVE STATISTICS WITH EXCEL Edward OMEY Brussels September 2014 Stormstraat 2, 1000 Brussels Belgium edward.omey@kuleuven.be 1. STARTING FROM DISCRETE RAW DATA Calculating the relevant statistics Frequency table Graphs The needle graph The EDF (discrete data) STARTING FROM CONTINUOUS RAW DATA The basic descriptive statistics The frequency table Graphs The histogram The EDF The box plot STARTING FROM GROUPED DATA Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 1

2 1. STARTING FROM DISCRETE RAW DATA We start from raw data such in TABLE 1. In Table1 we collected data concerning a discrete variable: the number of ill employees during 100 consecutive days. TABLE Before starting to make any calculations, it is better to put the data in 1 row or in 1 column. We choose to put the data in 1 column: Table 2. TABLE Calculating the relevant statistics To calculate the relevant DESCRIPTIVE STATISTICS, we activate Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 2

3 DATA DATA ANALYSIS We choose descriptive statistics. Saying OK gives: In the - INPUT RANGE: we select all data - GROUPED BY: we choose columns - OUTPUT RANGE: select one empty cell - SUMMARY STATISTICS: we only choose this We find: Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 3

4 Saying OK, we get: Column1 Mean 5,17 Standard Error 0, Median 5 Mode 4 Standard Deviation 2, Sample Variance 6, Kurtosis -0,65576 Skewness 0, Range 10 Minimum 1 Maximum 11 Sum 517 Count 100 the mean value this is s/ n this is s =sqrt(s²) this is s², the variance this characterizes the shape/peakedness this characterizes symmetry/skewness the sum of all data the number of data The smallest entry was 1 and the largest is equal to Frequency table Next we calculate the frequencies of the entries 1, 2,, 11. In EXCEL these entries are called the BINS. Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 4

5 We choose an empty column and manually enter the bins: BINS We select all cells next to the bins: Now we INSERT a FUNCTION by using the formulas and then insert function : Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 5

6 Choosing Insert function, we get the following screen: As CATEGORY, we choose STATISTICAL: Descriptive Statistics and EXCEL E. Omey BBA 6

7 Inside statistical we choose FREQUENCY: After OK, we get the following screen: - In DATA ARRAY: we select all our raw data - In BINS ARRAY: we select all the cells containing the bins Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 7

8 In the place of OK, now we use 3 buttons: CTRL and SHIFT and ENTER Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 8

9 As a result, we see the following next to the bins: BINS EXCEL has calculated that the frequency of the outcome 1 is equal to 3; the frequency of the outcome 9 is equal to 6 and so on. Now we can easily calculate relative frequencies (RELFREQ) and cumulative relative frequencies (CRF) We find RELFREQ by dividing the frequencies by n = 100. We find CRF by cumulating the relative frequencies: BINS freq relfreq CRF 1 3 0,03 0, ,12 0, ,11 0, ,2 0, ,2 0, ,03 0, ,1 0, ,08 0, ,06 0, ,06 0, , Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 9

10 1.3. Graphs The needle graph The NEEDLE graph is the graph of the relative frequencies. To make it we select the RELATIVE FREQUENCIES and then we choose insert and find different types of graphs: BINS F RF CRF 1 3 0,03 0, ,12 0, ,11 0, ,2 0, ,2 0, ,03 0, ,1 0, ,08 0, ,06 0, ,06 0, , We choose COLUMN and get a wide choice: Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 10

11 We select the first entry of 2D-COLUMN and we see Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 11

12 We remove the Series 1 and change the values on the horizontal axis by editing the horizontal axis First activate the horizontal axis Using the right side of the mouse we can select select data source and get: We edit the horizontal axis labels and choose the bins from the frequency table Then OK To make real needles, we click on one of the bars and use the right side of the mouse and choose Format Data Series to get the following form: Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 12

13 In this form we set the Gap Width as large as possible Descriptive Statistics and EXCEL E. Omey BBA 13

14 After closing, we get the following needle: Descriptive Statistics and EXCEL E. Omey BBA 14

15 rel freqencies Using the CHART TOOLS and LAYOUT, we can give the graph and the axis a name. We get the for example the following result needle graph absent employees Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 15

16 The EDF (discrete data) To construct the empirical distribution function, we select the CUMULATIVE RELATIVE FREQUENCIES and as in the previous paragraph, we insert a 2D-column We get: BINS F RF CRF 1 3 0,03 0, ,12 0, ,11 0, ,2 0, ,2 0, ,03 0, ,1 0, ,08 0, ,06 0, ,06 0, , Now we want the graph to be a STEPFUNCTION and we set the gap width equal to 0. We get: Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 16

17 We format the vertical axis and put the maximum value equal to 1: Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 17

18 Descriptive Statistics and EXCEL E. Omey BBA 18

19 Next we take care of the lay out and insert a title: The EDF This step-function is the EDF. Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 19

20 2. STARTING FROM CONTINUOUS RAW DATA We start from the following raw data (for convenience they are present here using columns) 1,16 2,77 17,54 4,82 0,55 0,19 0,13 5,73 14,12 7,13 0,99 20, ,7 0,62 11,79 0,48 4,64 6,94 2,27 2,55 1,04 0,56 0,72 1,56 13,65 9,48 1,36 3,11 0,06 3,45 32,37 10,54 0,25 10,19 0,45 22,57 6,81 16,18 5,59 7,88 4,73 10,25 4,31 2,16 5,08 3,69 3,51 0,79 1,36 7,95 19,46 19,94 14,12 3,49 4,92 12,52 4,5 0,85 10,88 3,52 0,24 4 3,1 0,81 3,14 3,94 7,23 3,55 4,71 4,74 2,92 7,11 6,33 0,46 7,06 6,03 9,65 2, ,35 17,27 2,17 4,96 25,24 18,99 3,45 5,9 4,09 19,11 3,6 9,66 5,06 8,6 0,2 3,53 3,22 29,07 0,51 0, The basic descriptive statistics As in the previous example, we take DATA DATA ANALYSIS DESCRIPTIVE STATISTICS: then we fill in the form: Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 20

21 and find: Column1 Mean 6,7627 Standard Error 0, Median 4,675 Mode 0,19 Standard Deviation 6, Sample Variance 46,21851 Kurtosis 2, Skewness 1, Range 32,31 Minimum 0,06 Maximum 32,37 Sum 676,27 Count 100 The smallest value is 0,06 and the largest is given by 32,37. The range is 32,37 0,06 = 32,31. Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 21

22 2.2. The frequency table We will construct a frequency table with about 10 classes of equal size. - the range is 32,31 - we choose 10 classes of size 32,31/10 = 3,231. we choose size = 3,2 - the lower bound of the first class (slightly smaller than the minimum) we choose 0. Now we construct a column with the bins as follows: BINS 0 3,2 6,4 9,6 12, ,2 22,4 25,6 28, ,2 Using the FREQUENCY FUNCTION as before, we find the following table: BINS 0 0 3,2 34 6,4 30 9, , ,2 6 22,4 3 25,6 2 28, ,2 1 Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 22

23 The sum (use the sum-symbol!) is given by 100. BINS 0 0 3,2 34 6,4 30 9, , ,2 6 22,4 3 25,6 2 28, , The table gives the following information - there are no observations less than or equal to 0 - there are 34 observations between 0 (not included) and 3.2 (included) - there are 30 observations between 3.2 (not included) and 6.4 (included) etc. We can calculate relative frequencies and cumulative relative frequencies as before: BINS AF RF CRF ,2 34 0,34 0,34 6,4 30 0,3 0,64 9,6 11 0,11 0,75 12,8 9 0,09 0, ,03 0,87 19,2 6 0,06 0,93 22,4 3 0,03 0,96 25,6 2 0,02 0,98 28, , ,01 0,99 35,2 1 0, Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 23

24 2.3. Graphs The histogram For the histogram, we select the relative frequencies, then INSERT and 2D Column. We find: We replace the number on the horizontal axis by the bins and remove the unnecessary series 1 To have a real histogram, we set the gap width equal to its minimal value 0. We find Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 24

25 The last step is to add a title. Histogram This is the histogram. Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 25

26 The EDF To construct the EDF in the continuous case, we select both the BINS ànd the CUM REL FREQ. Then we activate insert and choose scatter: BINS AF RF CRF ,2 34 0,34 0,34 6,4 30 0,3 0,64 9,6 11 0,11 0,75 12,8 9 0,09 0, ,03 0,87 19,2 6 0,06 0,93 22,4 3 0,03 0,96 25,6 2 0,02 0,98 28, , ,01 0,99 35,2 1 0, In Scatter we choose the second picture in the second line and we get: Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 26

27 We do some make up Remove series 1 Cutting the vertical axis at the value of 1 Adding a title The EDF We remove the series 1 and cut the vertical axis at 1. We get: This is the EDF. Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 27

28 3. The box plot The box plot is a graph in which we show - Q(0) = the minimum (the smallest observation) - Q(4) = the maximum (the largest observation) - Q(1) and Q(3): the 1 st and the 3 rd quartile. To calculate these quartiles from raw data we use the function QUARTILE. We get a form and to get the 1 st quartile, we enter 1 in the second bar of the table: We get Q(1) = 2,1675. The other quartiles can be calculated by replacing 1 by 0, or 3 or 4. Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 28

29 We get the following numbers: quartiles 0 0,06 1 2, , , ,37 To construct the box plot, we HAVE TO give the numbers in the following order: (you must include a date, any date) date Q(1) min max Q(3) 01/02/2011 2,1675 0,06 32,37 9,5225 Now we select both lines and INSERT OTHER CHARTS STOCK: Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 29

30 In STOCK we choose the 2 nd picture and then get the following result: Q(1) min max Q(3) We can do some esthetical steps, add a title and find the following graph: Box Plot max, 1, Q(3), 1, Q(1), 1, min, 1, 0.06 This is the box plot. Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 30

31 4. STARTING FROM GROUPED DATA In many cases we don t get the raw data, but the data are presented in a frequency table. To calculate the relevant statistics here, we use the built-in functions of EXCEL, cf We use the following example. Salary classes (x1000) AF ] 8,10] 5 ]10,12] 10 ]12,14] 30 ]14,16] 55 ]16,18] 35 ]18, 20] 29 ]20,22] 24 ]22,24] 7 total 195 The BINS here are the endpoints of the classes. In the terminology of BINS, we can make the following new table: BINS AF This table can be used to draw a histogram or to draw the EDF as before. To calculate some of the relevant descriptive statistics, in the frequency table we add a new column with the CLASS CENTRES. After this construction, we find: Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 31

32 Salary classes Class Centre AF ] 8,10] 9 5 ]10,12] ]12,14] ]14,16] ]16,18] ]18, 20] ]20,22] ]22,24] Now we can calculate the mean and the variance. The mean is given by Mean = (9 * * * 7)/195 = sumproduct(centres and frequencies)/ n The sumproduct function is available in the function wizard of EXCEL. we go to FOMULAS and INSERT FUNCTION In the mathematical functions we find the function SUMPRODUCT: Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 32

33 In the first array, we select the class centres, in the second array we select the absolute frequencies: After OK, we get the result: 3181 Of course, we need to divide this result by 195 to get the mean value 16,3128. To calculate the variance, we first have to calculate the second moment. Descriptive Statistics and EXCEL E. Omey BBA 33

34 The second moment is given by Second moment = (9² * ² * ² * 7)/195 = sumproduct (centers ànd centers ànd frequencies)/ n Again we use the sumproduct function, but now in the first array and in the second array, we select the class centers, in the 3 rd array we select the absolute frequencies. Now we get the following result: - We have to divide the result by 195 to find the second moment. Here we obtain the second moment = 276,5692 The variance is given by s² = second moment (mean)² = 276,5692 (16,3128)² = 10,46112 The standard deviation is given by the square root of the variance: s = sqrt(10,46112) = 3,234 To calculate other relevant parameters such as median and the quartiles, for GROUPED data, there are no special functions available in EXCEL. Descriptive Statistics and EXCEL E. Omey KUL@HUB BBA 34