**. **

**Let be the mean: =E[X], where E[X] denotes the expected value of X. **

**De nition. Gajendra. **

**e. **

**9 explains 81% of the variance. **

**The variance of a random variable X with expected value EX = is de ned as var(X) = E (X )2. Standard deviation: average distance from the mean. . **

**The variance should be regarded as (something like) the average of the diﬀerence of the actual values from the average. **

**There are many ways to quantify variability, however, here we will focus on the most common ones: variance, standard deviation, and coefficient of variation. As an important aside, in a normal distribution there is a specific relationship between the mean and SD: mean ± 1 SD includes 68. •r = 0. **

**. The standard deviation (the square root of variance) of a sample can be used to estimate a population's true variance. **

**Since the variance is measured in terms of x2,weoften wish to use the standard deviation where σ = √ variance. **

**1. **

**. We now consider the standard deviation, which we know is de ned as sd(X) = p var(X) for a random variable X. **

**Square each. We now consider the standard deviation, which we know is de ned as sd(X) = p var(X) for a random variable X. **

**Mean Estimator The uniformly minimum**

**variance**unbiased (UMVU) es-timator of is #"[1, p.**•"R-squared" is a standard way of measuring the proportion of variance we can explain in one variable using one or more other variables. **

**Step 2: For each data point, find the square of its distance to the mean. **

**Suppose Z = h(X,Y), where X is the sample mean of measured values of X, and likewise for Y. . ·. **

**The formula is: µ d e v i a t i o n s c o r e = x − µ. . 11=5. In order to understand the differences between these two observations of statistical spread, one must first understand what each represents: Variance represents all data points in a set and is calculated by. . With large enough samples, the difference is small. **

**1. **

**If f(x i) is the probability distribution function for a random variable with range fx 1;x 2;x 3;:::gand mean = E(X) then: Var(X) = ˙2 = (x 1. estimators of the mean, variance, and standard deviation. **

**To have a good understanding of these, it is. **

**These differences are called deviations. **

**Variance** & **Standard Deviation** If we model a factor as a random variable with a specified probability distribution, then the **variance** of the factor is the expectation, or mean, of the squared **deviation** of the factor from its expected value or mean.

**The only difference is the squaring of the distances. **

**e. **