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DESCRIPTIVE-STATISTICS.pptxxxxxxcxxxcxdff
- 1. ADVANCE STATISTICS Jemer M.Mabazza, LLB, PhD Graduate School Mallig Plains Colleges
- 2. DESCRIPTIVE STATISTICS DescriptiveStatistics • Two main function of descriptive statistics 1. To carry out the measure of central tendency 2. To calculate a measure of dispersion or variability • Measure of Central Tendency 1. Mean. Refers to descriptive statistic that summarizes the centre of a range of values. It is computed by mean of adding all the scores or values in each condition and dividing them by the number of scores or values. 2. Median. Refers to descriptive statistic that summarizes the middle of a range of values after arranging items from highest to lowest or vice versa. 3. Mode. Refers to the descriptive statistics that identifies the value with the higher frequency.
- 3. 10 13 1322 23 25 30 35 40 • Example Mean = 10+ 13+13+22+23+25+30+35+40 9 = 211 9 = 23.44 Median = 23 • RULES: If N is an ODD NUMBERS, get the EXACT MIDDLE SCORE/NUMBER If N is an EVEN NUMBERS, get the AVERAGE OF TWO MIDDLE SCORES/NUMBERS Mode = 13 DESCRIPTIVE STATISTICS
- 4. • Illustration ofthe Rules in the Median If N is an ODD NUMBERS, get the EXACT MIDDLE SCORE/NUMBER ODD NUMBERS: 5 10 15 Median = 10 If N is an EVEN NUMBERS, get the AVERAGE OF TWO MIDDLE SCORES/NUMBERS EVEN NUMBERS: 2 4 6 8 Median = 4 + 6 2 Median = 5 DESCRIPTIVE STATISTICS
- 5. Mean (forUngrouped Data): Example FORMULA: X = ΣX N X = (2+4+6+8+10) 5 X = 30 5 X = 6 Mean (Grouped Data): Example FORMULA: X = ΣfX N Class Interval 118-126 127-135 136-144 145-153 154-162 f 3 5 9 12 5 _______ 34 X 122 131 140 149 158 fX 366 655 1260 1788 790 _______ 4859 X = 4859 34 X = 142.9 DESCRIPTIVE STATISTICS
- 6. • Measure ofDispersion 1. Range. Refers simply the highest value minus the lowest value FORMULA: Range ( R ) = Highest Score (HS) – Lowest Score (LS) EXAMPLE: 50 60 10 25 25 30 20 40 R = HS – LS = 60 – 10 = 50 2 3 4 5 6 7 8 9 R = HS – LS = 9 – 2 = 7 DESCRIPTIVE STATISTICS
- 7. 2. Variance. Thisconcerned with the extent to which the values in a data set differ from the mean. • Sample Variance. Describes how far the sample scores are spread out around the sample mean. FORMULA: SX 2 = Σ𝑿𝟐 − 𝚺𝑿 𝟐/𝑵 𝑵 Where: SX 2 = sample variance ΣX = sum of the scores 𝚺𝑿 𝟐 = square sum of the scores N = number of scores in the sample DESCRIPTIVE STATISTICS
- 8. EXAMPLE: 6, 7,8, 9, 10, 11, 12, 13, 14, 15 SX 2 = 𝟔𝟐+ 𝟕𝟐+ 𝟖𝟐+ 𝟗𝟐+ 𝟏𝟎𝟐+ 𝟏𝟏𝟐+ 𝟏𝟐𝟐+ 𝟏𝟑𝟐+ 𝟏𝟒𝟐+ 𝟏𝟓𝟐 − 𝟔+𝟕+𝟖+𝟗+𝟏𝟎+𝟏𝟏+𝟏𝟐+𝟏𝟑+𝟏𝟒+𝟏𝟓 𝟐 /𝟏𝟎 𝟏𝟎 SX 2 = 𝟏, 𝟏𝟖𝟓− 𝟏𝟎𝟓 𝟐 /𝟏𝟎 𝟏𝟎 SX 2 = 𝟏, 𝟏𝟖𝟓−𝟏𝟏,𝟎𝟐𝟓/𝟏𝟎 𝟏𝟎 SX 2 = 𝟏, 𝟏𝟖𝟓−𝟏,𝟏𝟎𝟐.𝟓 𝟏𝟎 SX 2 = 𝟖𝟐.𝟓 𝟏𝟎 SX 2 = 8.25 DESCRIPTIVE STATISTICS
- 9. • Estimated PopulationVariance. This is an estimate of how far the scores in the population would spread out around the population mean. FORMULA: SX 2 = Σ𝑿𝟐 − 𝚺𝑿 𝟐/𝑵 𝑵 −𝟏 Where: SX 2 = sample variance ΣX = sum of the scores 𝚺𝑿 𝟐 = square sum of the scores N – 1 = number of scores in the sample minus 1 DESCRIPTIVE STATISTICS
- 10. EXAMPLE: 6, 7,8, 9, 10, 11, 12, 13, 14, 15 SX 2 = 𝟔𝟐+ 𝟕𝟐+ 𝟖𝟐+ 𝟗𝟐+ 𝟏𝟎𝟐+ 𝟏𝟏𝟐+ 𝟏𝟐𝟐+ 𝟏𝟑𝟐+ 𝟏𝟒𝟐+ 𝟏𝟓𝟐 − 𝟔+𝟕+𝟖+𝟗+𝟏𝟎+𝟏𝟏+𝟏𝟐+𝟏𝟑+𝟏𝟒+𝟏𝟓 𝟐 /𝟏𝟎 𝟏𝟎 −𝟏 SX 2 = 𝟏, 𝟏𝟖𝟓− 𝟏𝟎𝟓 𝟐 /𝟏𝟎 𝟏𝟎 −𝟏 SX 2 = 𝟏, 𝟏𝟖𝟓−𝟏𝟏,𝟎𝟐𝟓/𝟏𝟎 𝟏𝟎 −𝟏 SX 2 = 𝟏, 𝟏𝟖𝟓−𝟏,𝟏𝟎𝟐.𝟓 𝟏𝟎 −𝟏 SX 2 = 𝟖𝟐.𝟓 𝟏𝟎 −𝟏 SX 2 = 𝟖𝟐.𝟓 𝟗 SX 2 =9.17 DESCRIPTIVE STATISTICS
- 11. 3. Standard Deviation.Refers to the positive square root of the arithmetic mean of the squared deviations from the mean of the distribution EQUATION: S2 = 𝑖=1 𝑁 (𝑋1 −X) 2 𝑁 Where: (𝑿 − X)𝟐 = sum of squares of the differences between scores and the mean N = number of scores in the sample From the equation, we can simply say that standard deviation is the square root of the variance, and the variance is the square of the standard deviation DESCRIPTIVE STATISTICS
- 12. • Computation ofthe Standard Deviation for Ungrouped Data FORMULA: S = Σ(𝑋−X)2 𝑁 • EXAMPLE X 15 15 17 18 20 85 𝑋 = 85 5 = 17 (X - 𝑿) - 2 - 2 - 0 1 3 (𝑋 − X)2 4 4 0 1 9 18 S= 18 5 = 3.6 = 1.9 Step 1: Compute the mean Step 2: Get the deviations from the mean Step 3: Square the deviations from mean Step 4: Get the sum of the squared deviation Step 5: Divide the sum by the total frequency Step 5: Extract the square root of the quotient DESCRIPTIVE STATISTICS
- 13. • Computation ofthe Standard Deviation for Grouped Data 1. Long Method. This method makes use of the following formula: FORMULA: s = Σ𝑓(X2) N−1 − (Σ 𝑓𝑥)2 𝑁 (𝑁−1) 2. Coded Deviation Method FORMULA: s= 𝑖 Σ 𝑓(𝑑′)2 𝑁 −1 − (Σ 𝑓𝑑′)2 𝑁 (𝑁 −1) d′ = X − X′ 𝑖 FORMULA for CODED DEVIATION FROM THE MEAN: DESCRIPTIVE STATISTICS
- 14. DESCRIPTIVE STATISTICS • EXAMPLE:RAW DATA 120 140 161 148 130 156 165 146 142 175 133 150 149 139 143 151 138 150 158 148 180 170 124 161 137 128 147 149 152 142 138 153 168 142 147 118 167 129 130 159
- 15. 1. Long Method.EXAMPLE Class Interval 118-126 127-135 136-144 145-153 154-162 163-171 172-180 f 3 5 9 12 5 4 2 _______ 40 X 122 131 140 149 158 167 176 fX 366 655 1260 1788 790 668 352 𝐗 𝟐 14884 17161 19600 22201 24964 27889 30976 𝒇𝐗 𝟐 44652 85805 176400 266412 124820 111556 61952 _______ 871597 _______ 5879 s = Σ𝑓(X2) N−1 − (Σ 𝑓𝑥)2 𝑁 (𝑁−1) s = 871597 39 − 58792 40 (39) s = 22348.64 − 34562641 1560 s = 22348.64 − 22155.54 s = 22348.64 − 22155.54 s = 193.1 s = 13.8 DESCRIPTIVE STATISTICS
- 16. 2. Coded DeviationMethod. EXAMPLE Class Interval 118-126 127-135 136-144 145-153 154-162 163-171 172-180 f 3 5 9 12 5 4 2 _______ 40 X 122 131 140 149 158 167 176 d’ -3 -2 -1 0 1 2 3 𝒇𝒅′ -9 -10 -9 0 5 8 6 (𝐝′) 𝟐 9 4 1 0 1 4 9 _______ -9 f(𝐝′) 𝟐 27 20 9 0 5 16 18 _______ 95 s= 𝑖 Σ 𝑓(𝑑′)2 𝑁 −1 − (Σ 𝑓𝑑′)2 𝑁 (𝑁 −1) s= 9 95 39 − (−9)2 40 (39) s= 9 2.44 − 0.05 s= 9 2.39 s= 9 (1.55) s= 13.95 DESCRIPTIVE STATISTICS
Editor's Notes
- #3 NOTE 1: (Descriptive Statistics). Refers to the statistical procedure used in describing the properties of samples, or of population where complete population data are available. NOTE 2: (Mean). Simply mean the sum of all items or terms divided by the total number of items or terms. NOTE 3 (Median). Simply mean the value of the middle term after arranging the data in an ascending or descending order.
- #7 NOTE 1 (UNGROUPED DATA): For ungrouped data, it means the data that is not arranged in a frequency distribution.
- #12 NOTE 1 (Standard Deviation). It is a statistical tool to determine the homogeneity or similarity of degree or dimension of given variables or the heterogeneity or degree of dispersal of variables.
- #14 NOTE 1 (LONG METHOD). This is more convenient to use than the first method. However, when the class marks have very big values, this may not be practical too. This method is used preferably when the values of the class marks are relatively small. NOTE 2 (CODED DEVIATION METHOD) This method is the most convenient among the three methods for computing the standard deviation from grouped data.
- #16 NOTE: (1) To determine the range of the class interval, get the difference between the highest and lowest values in the set of data. The range therefore is 175-118=57. (2) Determine the number of class intervals or categories desired. The ideal number of class interval is somewhere between 5 and 15. (3) If the desired number of class interval is seven, then the size of the class interval (symbolized by i) is 57 / 7 =8.13. Here, we may use 8 or 9 as class interval. (4) Write the class intervals starting with the lowest lower limit, then add the value of i. NOTE: (1) Compute the class mark by adding the lower and upper limit of the class interval, then dividing the sum by 2. The class mark is the representative value of the corresponding interval
- #17 NOTE: To get the coded deviation from the mean (d’) divide the deviation of the class marks from the assumed mean by the size of the class interval d ′ = X − X ′ 𝑖 NOTE: EXAMPLE: (122-149)/8 = -3