CHAPTER 4_ MATH IN THE MODERN WORLD.pptx

STATISTICS AND DATA MANAGEMENT

1. CHAPTER 4: STATISTICS (Data Management) PREPARED BY: JOHN LUIS M. BANTOLINO, LPT TOPIC 3: MEASURES OF DISPERSION

2. TOPIC 3: MEASURES OF DISPERSION PREPARED BY: JOHN LUIS M. BANTOLINO, LPT VARIANCE The variance of a population is equal to the sum of the squared deviations about the mean divided by the number of scores. STANDARD DEVIATION AND VARIANCE STANDARD DEVIATION The standard deviation is equal to the square root of the variance. If the value are clustered tightly about their mean, the standard deviation is small and if the value become more scattered about their mean, the standard deviation for these sets is large.

3. Computation of Variance and Standard Deviation for Ungrouped Data The variance of a population ( ) and population standard deviation (σ) for ungrouped data can be computed from the formula:   Variance of a Population (Ungrouped Data) = Standard Deviation of a Population (Ungrouped Data) σ =   Where: = variance of a population σ = population standard deviation x = values observations in the population = population mean N = total number of observations in the population   TOPIC 3: MEASURES OF DISPERSION PREPARED BY: JOHN LUIS M. BANTOLINO, LPT

4. Computation of Variance and Standard Deviation for Ungrouped Data The sample variance (s2) and sample standard deviation (s) for ungrouped data can be computed from the formula : Variance of a Sample (Ungrouped Data) = Standard Deviation of a Sample (Ungrouped Data) =   Where: = variance of a sample = sample standard deviation x = values observations in the sample = sample mean n = total number of observations in the sample   TOPIC 3: MEASURES OF DISPERSION PREPARED BY: JOHN LUIS M. BANTOLINO, LPT

5. TOPIC 3: MEASURES OF DISPERSION PREPARED BY: JOHN LUIS M. BANTOLINO, LPT For instance: Let us use the data of Company B in our previous lesson in Mean Deviation. Since the given data is a sample and the data were ungrouped, we will use the formula for sample variance and sample standard deviation for ungrouped data. =Σ(x- x)2 /n-1→ = 1000/(5-1)→ = 1000/(4)→ = 250   Variance of a Sample (Ungrouped Data) Standard Deviation of a Sample (Ungrouped Data) s = Σ(x- x)2 /n-1 → 1000/(5-1) → 250 → s = 15.81

6. Computation of Variance and Standard Deviation for G rouped Data The variance of a population ( ) and population standard deviation (σ) for g rouped data can be computed from the formula:   Variance of a Population ( G rouped Data) = Standard Deviation of a Population ( G rouped Data) σ =   Where: = variance of a population σ = population standard deviation = class mark f = frequency = population mean N = total number of observations in the population   TOPIC 3: MEASURES OF DISPERSION PREPARED BY: JOHN LUIS M. BANTOLINO, LPT

7. Computation of Variance and Standard Deviation for Ungrouped Data The sample variance (s2) and sample standard deviation (s) for grouped data can be computed from the formula : Variance of a Sample (Ungrouped Data) = Standard Deviation of a Sample (Ungrouped Data) =   Where: = variance of a sample = sample standard deviation = class mark f = frequency = sample mean n = total number of observations in the sample   TOPIC 3: MEASURES OF DISPERSION PREPARED BY: JOHN LUIS M. BANTOLINO, LPT

8. TOPIC 3: MEASURES OF DISPERSION PREPARED BY: JOHN LUIS M. BANTOLINO, LPT Let us use again the grouped data of the scores of students in an exam that we organized beforehand in the previous lessons and also consider it as a sample (not a population anymore). We knew already that its mean is 83.   Variance of a Sample (Ungrouped Data) Standard Deviation of a Sample (Ungrouped Data) s= Σf(X- x)2 /n-1 → 2600/(50-1) → 53.06122… → s= 7.28 With a variance of 53.06,the standard deviation is7.28. The whole data set can be can be interpreted that the scores of the students are between 104.84 and 61.16.