S05
Statistics 305
Scalar Numerical Summary Measures
To estimate the parameters in a population, which is not one of our theoretical populations for
which parameters are known, we take a random sample of size n from the population and
compute scalar numerical summary values. For example, the mean of sample elements is the
estimate of population mean µ. Such scalar numerical measures computed from sample data are
called Statistics. Thus the sample mean, sample variance, and quantiles computed from the
sample are all statistics which estimate the corresponding parameters of the sampled population.
The formulas for computing the statistics sample mean, sample variance and range are shown
below. We gave textbook and JMP methods for computing quantiles in a sample in a previous
handout. Such quantiles are, of course, statistics. Mathematical theory gives general expressions
for finding parameters in populations. These expressions are shown below for population mean
and variance. You can see that the density function f(x) is involved. The derivation of the
specific expressions for µ and σ 2 in the theoretical populations discussed earlier is not easy and
is not a part of Stat 305. Handout 4 gave the results of such derivations.
From Samples
x1 ≤ x 2 ≤ K ≤ x n . The following are Statistics.
1. Sample mean x =
1 n
∑ xi
n i=1
2. Median Q(0.5) computed from the finite dataset using JMP or textbook definition.
3. Quantiles Q(p) computed from the finite dataset using JMP or textbook definition.
4. Range R = x n – x 1
5. Sample variance s 2 =
1 n
(x i − x ) 2
∑
n − 1 i =1
6. Sample standard deviation s = s 2
1
S05
For Populations – The following are general expressions for Parameters, which are used to
derive the parameters in Handout 4.
Discrete Distributions
Continuous Distributions
1.
Mean µ = Σ xf (x )
µ=
2.
Median Q(0.5)
Q(0.5)
3.
Quantiles Q(p)
Q(p)
4.
Variance σ 2 = Σ ( x − µ ) 2 f ( x )
σ 2 = ∫ ( x − µ ) 2 f ( x ) dx
5.
Standard Devia tion σ = σ 2
σ = σ2
( f (x ) is the Density function).
2
∫
xf ( x ) dx