3-1-4 Statistics

The observation table gives Number 1 through 10 and Score 75, 88, 95, 65, 75, 62, 86, 72, 62, 75. The frequency table columns are Class and Frequency: 60 to 69 has 3; 70 to 79 has 4; 80 to 89 has 2; 90 to 100 has 1. The histogram uses vertical People labels 0, 1, 2, 3, 4 and horizontal Score labels 60, 70, 80, 90, 100. All three original panels are kept together.

[Observation Values] Number 1 2 3 4 5 6 7 8 9 10 Score 75 88 95 65 75 62 86 72 62 75 [Frequency distribution table] [Histogram] Class Frequency 60 to 69 3 70 to 79 4 80 to 89 2 90 to 100 1

The observation values are 60, 62, 62, 62, 64, 66, 66, 68, 70, 70. The original results are Average 65, Mode 62 with 3 occurrences, Median 65 and Range 10. The printed division signs and source minus character are retained.

Examples: Calculate the representative values for the following observation values. Observation Values 60 62 62 62 64 66 66 68 70 70 • Average = (60 + 62 + 62 +62 + 64 + 66 + 66 + 68 + 70 + 70) ÷ 10 = 65 • Mode = 62 (62 appears the most in the set with 3 occurrences.) • Median = (64 + 66) ÷ 2 = 65 • Range = 70 − 60 = 10

The rows are Variance and Standard deviation. Variance is the printed capital-Sigma expression for squared deviations divided by the number of observation values; its note states Sigma means summation. Standard deviation is the square root of variance. The original deviation-value paragraph uses average beta and standard deviation alpha, with example alpha = 10, beta = 50. Its fraction has Observation value minus Average above Standard deviation, then times alpha plus beta.

Name Meaning Σ(Deviation of observation values from the average)² ÷ Number of Variance observation values Note: Σ means summation. The square root ( ) of the variance Deviation value can be calculated from the standard deviation by using the expression below. The deviation value is a value which is standardized so that the average can be β, and the standard deviation Standard can be α. In such as exams, these deviation values are commonly deviation written as, for example, α = 10, β = 50. Deviation value = ^{Observation value − Average} × α + β Standard deviation

The X observations are 74, 76, 78, 80, 80, 80, 82, 82, 84, 84. The Y observation row is 60, 66, 72, 78, 80, 84, 88, 90, 90, 92. The original wording states an average of 80 for both. Each original calculation and displayed result is transcribed below in its printed order.

Examples: Calculate the variance and standard deviation from two sets of observation values, both of which have an average of 80. Observation values X 74 76 78 80 80 80 82 82 84 84 Observation values Y 60 66 72 78 80 84 88 90 90 92

The original correlation-analysis box continues onto the next page. It states coefficients from 1 to -1 and describes the direction and strength. The left scatter plot is captioned Strong positive correlation. The right is captioned Weak negative correlation. The source captions are retained unchanged.

• Correlation analysis Correlation analysis is a method of statistical analysis which uses a correlation coefficient to find the degree of correlation between observation values. Correlation coefficients range between 1 and −1. The closer the correlation coefficient is to 1, the stronger the positive correlation. The closer the correlation coefficient is to −1, the stronger the negative correlation. The closer to 0, the weaker the correlation.

The continued box contains the original Regression Analysis paragraph and graph. The line is labelled Regression line. The axes are x and y, the intercept is b, and the callout gives y = a x + b, a: slope, b: y-intercept.

• Regression Analysis Regression analysis is a statistical technique for estimating one of data x and y, which are correlated with each other, from the other. In particular, the regression model is called a regression line, which is represented by the linear equation y = a x + b.

The original panels are Normal distribution, Poisson distribution and Exponential distribution. Their source descriptions identify continuous, discrete and continuous probability variables respectively. The original drawings are a bell curve, bars and a decreasing curve. No distribution equation has been added.

• Normal distribution • Poisson distribution • Exponential distribution This is a probability This is a probability This is a probability distribution for distribution for discrete distribution for continuous continuous probability probability variables. probability variables. variables.

The original graph has vertical f(x), horizontal x and a marked 0. The center is mu. Two sigma-marked intervals extend to either side of mu, with dashed vertical lines. All original labels and the curve are retained.

The original part-length example gives average mu 3.2 cm and standard deviation sigma 0.1 cm, rejecting lengths less than 3.0 cm. The source graph is labelled u and p. The standard normal distribution table gives (u,p): (0.0,0.500), (0.5,0.309), (1.0,0.159), (1.5,0.067), (2.0,0.023), (2.5,0.006), (3.0,0.001). Absolute-value bars enclose each numerator of the original normalization calculation. The example continues on the next page.

Examples: The following is a normal distribution which represents the length of a part that is produced at a certain factory. • Average μ: 3.2 cm • Standard deviation σ: 0.1 cm This part will fail inspection if it is less than 3.0 cm in length. Calculate what percentage of all manufactured parts that will fail inspection. Standard normal distribution table u p 0.0 0.500 0.5 0.309 1.0 0.159 1.5 0.067 2.0 0.023 2.5 0.006 3.0 0.001 1) From the normal distribution N(3.2, 0.1²), normalize the length x = 3.0 at which the part fails inspection to the deviation value u of the standard normal distribution N(0, 1²).

The original continuation selects p = 0.023 at u = 2.0 and concludes 2.3% of products fails inspection. Original wording is retained.

2) Find the probability p where u = 2.0 from the standard normal distribution table. p = 0.023 => 2.3% of products fails inspection.