x̄ - > standard deviation | Student's t distribution | degrees of freedom | ν = 17

 

Standard Deviation, Student's t Distribution, Degrees of Freedom

Standard Deviation is a measure of the amount of variation or dispersion of a set of values from their mean. It is calculated as the square root of the variance, which is the average of the squared differences from the mean. Standard deviation is a commonly used measure of the spread of data in statistics, and is often used in hypothesis testing and confidence interval calculations.

Student's t Distribution is a probability distribution that arises in statistics when the sample size is small and the population standard deviation is unknown. The t distribution is similar in shape to the normal distribution but has heavier tails, which means it has more probability in the tails than the normal distribution. The t distribution is used in hypothesis testing, confidence interval calculations, and other statistical analyses.

Degrees of Freedom is a parameter that determines the shape of the t distribution. It is defined as the number of independent pieces of information that are used to estimate a parameter or test a hypothesis. For example, in a two-sample t-test, the degrees of freedom is equal to the sum of the sample sizes minus two. The degrees of freedom affects the width of the t distribution, with larger degrees of freedom leading to a narrower distribution.

If the degrees of freedom is given as ν = 17, this means that the t distribution being referred to has 17 degrees of freedom. This value is typically used in statistical calculations, such as calculating confidence intervals or performing hypothesis tests, where the sample size is small and the population standard deviation is unknown.

In summary, standard deviation is a measure of the spread of data, the t distribution is a probability distribution used in statistical analyses, and degrees of freedom determines the shape of the t distribution. Together, these concepts are important in many areas of statistics and can be used to make informed decisions based on data.

sqrt(17/15)≈1.06458

mean | 0

mode | 0

standard deviation | sqrt(17/15)≈1.06458

variance | 17/15≈1.13333

skewness | 0

(228581619826688 sqrt(17))/(6435 π (x^2 + 17)^9)

P (X<=x) = piecewise | 1/2 I_(17/(x^2 + 17))(17/2, 1/2) | x<=0

1/2 (I_(x^2/(x^2 + 17))(1/2, 17/2) + 1) | (otherwise)

10th | -1.33338

25th | -0.689195

50th | 0

75th | 0.689195

90th | 1.33338