Download Frequency Distributions and Measures of Central

Chapter 10
Introduction to Statistics
10.1 Frequency Distributions;
Measures Of Central Tendency
•
•
•
•
Grouped frequency distribution
Histogram, Frequency polygon
Stem and leaf
Summation notation (sigma
notation)
n
x1 + x2 + x3+ ….+ xn =  xi
i 1
• The mean (arithmetic everage)
The mean of the n numbers x1, x2, x3, … xn is
x1  x2  ...  xn  x

x
n
n
MEAN OF A GROUPED DISTRIBUTION
• The mean of a distribution where x represents
the midpoints, f the frequencies, and n=  f , is
 ( xf )
x
n
• Median: The middle entry in a set of data
arranged in either increasing or decreasing order.
If there is an even number of entries, the median
is defined to be the mean of the two center
entries.
• Mode: the most frequent entry. If each entry has
the same frequency, there is no mode.
10.2 MEASURES OF VARIATION
• Range of a list of numbers: max – min
• Deviations from the mean of a sample of n
numbers x1, x2 , x3, … xn, with mean x is:
x1 – x
x2 – x
…
xn –
x
10.2 MEASURES OF VARIATION
SAMPLE VARIANCE
• The variance of a sample of n numbers x1,
x2 , x3, … xn, with mean
s2 =
 ( x  x)
n 1
2
Population variance: s2 =
x , is
 ( x  x)
n
2
10.2 MEASURES OF VARIATION
SAMPLE STANDARD DEVIATION
• The standard deviation of n numbers
x1, x2 , x3, …, xn, with mean x , is
s
( x  x )
n 1
2
10.2 MEASURES OF VARIATION
STANDARD DEVIATION FOR A GROUP
DISTRIBUTION
•The standard deviation for a distribution
with mean x, where x is an interval
midpoint with frequency f, and n =  f , is
s
 fx
2
 nx
n 1
2
10.3 NORMAL DISTRIBUTION
• Continuous distribution
Outcome can take any real number
• Skewed distribution
The peak is not at the center
• Normal distribution
bell-shaped curve (4 basic properties)
• Normal curves
The graph of normal distribution
4 Basic Properties Of
Normal Distribution
1. The peak occurs directly above the mean
2. The curve is symetric about the vertical
line through the mean.
3. The curve never touches the x-axis
4. The area under the curve is 1
The mean: m
Standard deviation: s
Standard normal curve: m0, s  1
AREA UNDER NORMAL CURVE
There area of the shaded region under the
normal curve from a to b is the probability
that an observed data value will be between
a and b.
Z-score
• z-score
• If a normal distribution has mean m and
standard deviation s, then the z-score for
the number is
xm
z=
s
AREA UNDER NORMAL CURVE
The area under normal curve between x=a
and x=b is the same as the area under the
standard normal curve between the zscore for a and the z-score for b.
10.4 Normal Approximation to
the Binomial Distribution
• The expected number of successes in n
binomial trials is np, where p is the
probability of success in a single trial.
• Variance and standard deviation
s2 = np(1 – p) and s =
np(1 p)
• Suppose an experiment is a series of n
independent trials, where the probability of
a success in a single trial is always p. Let
x be the number of successes in the n
trials. Then the probability that exactly x
success will occur in n trials is given by:
n x
n x
  p (1  p)
 x