Download The standard error of the sample mean and

The standard error of the sample
mean and confidence intervals
How far is the average sample mean
from the population mean?
In what interval around mu can we
expect to find 95% or 99% or sample
means
An introduction to random samples
• When we speak about samples in statistics,
we are talking about random samples.
• Random samples are samples that are
obtained in line with very specific rules.
• If those rules are followed, the sample will
be representative of the population from
which it is drawn.
Random samples: Some principles
• In a random sample, each and every score must
have an equal chance of being chosen each time
you add a score to the sample.
• Thus, the same score can be selected more than once,
simply by chance. (This is called sampling with
replacement.)
• The number of scores in a sample is called “n.”
(Small n, not capital N.)
• Sample statistics based on random samples provide
least squared, unbiased estimates of their population
parameters.
The first way a random sample is
representative of its population
• One way a random sample will be
representative of the population is that the
sample mean will be a good estimate of the
population mean.
• Sample means are better estimates of mu
than are individual scores.
• Thus, on the average, sample means are
closer to mu than are individual scores.
The variance and the standard deviation
are the basis for the rest of this chapter.
• In Chapter 1 you learned to compute the
average squared distance of individual
scores from mu. We called it the variance.
• Taking a square root, you got the standard
deviation.
• Now we are going to ask a slightly different
question and transform the variance and
standard deviation in another way.
As you add scores to a random
sample
• Each randomly selected score tends to correct the
sample mean back toward mu
• If we have several samples drawn from a single
population, as we add scores to each sample, each
sample mean gets closer to mu.
• Since the sample means are all getting closer to
mu, they will also be getting closer to each other.
As you add scores to a random
sample – larger vs. smaller
samples
• The larger the random samples, the closer
their means will be to mu, on the average.
• Therefore: The larger the random samples,
the closer their means will be to each other,
on the average.
Let’s see how that happens
Population is 1320 students taking a test.
 is 72.00,  = 12
Let’s randomly sample one student at a time and see what
happens.We’ll create a random sample with 8 students’ scores in the
sample.
Test Scores
F
r
e
q
u
e
n
c
y
Standard
deviations
Scores
Mean
score
3
2
1
0
1
2
3
36
48
60
72
84
96
108
Sample scores:
102
Means:
72
87
66
80
76
79
66
76.4
78
69
76.7 75.6
63
74.0
How much closer to mu does the sample mean get
when you increase n, the size of the sample? (1)
• The average squared distance of individual
scores from mu, the population mean is
called the variance. You learned to compute
it in Chapter 1.
• The symbol for the mean of a sample is the
letter X with a bar over it.We will write that
as X-bar or X .
How much closer to mu does the sample mean get
when you increase n, the size of the sample? (2)
• The average squared distance of sample
means from mu is the average squared
distance of individual scores from mu
divided by n, the size of the sample.
• Let’s put that in a formula
• sigma2X-bar = sigma2/n
Then take a square root to obtain
the standard error of the mean,
the average unsquared distance of
sample means from mu.
sigma X  sigma / n
• This is important. So here are several
definitions of the standard error of the mean.
They are all correct, but ignore any that
confuse you!
The standard error of the sample mean
• As you know, the square root of the variance is
called the standard deviation. It is the average
unsquared distance of individual scores from mu.
• The average unsquared distance of sample means
from mu is the square root of sigma2X-bar
• The square root of sigma2X-bar = sigmaX-bar.
sigmaX-bar is called the standard error of the sample
mean or, more briefly, the standard error of the
mean. Here are the formulae
sigma2X-bar = sigma2/n
(Then, to get the standard error of the mean take a
square root of sigma2/n).
sigmaX-bar = sigma/ n
The standard error of the mean
• Let’s translate the formula into English, just
to be sure you understand it. Here is the
formula again: sigmaX-bar = sigma/ n
• In English: The average (unsquared)
distance of sample means from mu equals
the ordinary standard deviation divided by
the square root of the sample size.
Still another way to say the same thing:
The standard error of the mean is the
standard deviation of the means of
random samples of a specific size (n).
This last definition sometimes confuses
people. If it confuses you, just
remember:
The standard error of the mean is the
averaged unsquared distance of sample
means from mu.
Let’s check and make sure that the
formula is correct. Let’s see that the
standard error equals the ordinary
standard deviation divided by the square
root of n. To do that, let’s start with a
tiny population: N=5
• Here are all the scores in a population: 1,3,5,7,9.
• The scores in this population form a perfectly
rectangular distribution.
• Mu = 5.00
Figur e 4.5: Scores of the 5 research pa rticipants in this population.
Frequency
3
2
1
x
1.00
x
2.00
3.00
x
4.00
5.00
x
6.00
7.00
x
8.00
9.00
Computing sigma
• SS=(1-5)2+(3-5)2+(5-5)2+ (7-5)2+ (9-5)2=40
• sigma2=SS/N=40/5=8.00
• sigma = 2.83
If we did compute a standard deviation
of sample means from mu, it should give
the same result as the formula
• Let’s see if it does.
• We can only do all the computations if we have a very
small population and an even tinier sample.
• Let’s use the population of 5 scores we just looked at
(sigma = 2.83). We’ll look at all the samples with n = 2.
• If the formula is right, the average unsquared distance of
sample means, n=2, should be 2.83/ 2= 2.83/1.414 = 2.00.
• Is that right? To find out, let’s compute the standard error
of the mean from the differences between the means of all
possible samples (n=2) from mu.
Table 4.10: List of all 25 possible samples (n=2) of scores from the tiny population of
five scores (1, 3, 5, 7, & 9 ) and direct computation of the standard
error of these means. In this case, the standard error is computed from these means
(n=2) just as we would the standard deviation of an ordinary set of scores (n=1).
Sample Scores
AA
AB
AC
AD
AE
BA
BB
BC
BD
BE
CA
CB
CC
CD
CE
1,1
1,3
1,5
1,7
1,9
3,1
3,3
3,5
3,7
3,9
5,1
5,3
5,5
5,7
5,9
X
1.00
2.00
3.00
4.00
5.00
2.00
3.00
4.00
5.00
6.00
3.00
4.00
5.00
6.00
7.00
Sample Scores
DA
DB
DC
DD
DE
EA
EB
EC
ED
EE
7,1
7,3
7,5
7,7
7,9
9,1
9,3
9,5
9,7
9,9
X
4.00
5.00
6.00
7.00
8.00
5.00
6.00
7.00
8.00
9.00
Summary statistics
(all samples, n=2)
S X = 125.00
N = 25
mu = 5.00
SS X = 100.00
sigma X 2 = 4.00
sigma X = 2.00
Means of all possible 25 samples (n=2) from this population of scores 1. 3, 5, 7. and 9.
Frequency
5
4
3
2
1
Score
X
1.00
X
X
2.00
X
X
X
3.00
X
X
X
X
4.00
X
X
X
X
X
5.00
X
X
X
X
6.00
X
X
X
7.00
X
X
8.00
X
9.00
Table 4.10: List of all 25 possible samples (n=2) of scores from the tiny population of
five scores (1, 3, 5, 7, & 9 ) and direct computation of the standard
error of these means. In this case, the standard error is computed from these means
(n=2) just as we would the standard deviation of an ordinary set of scores (n=1).
Sample Scores
AA
AB
AC
AD
AE
BA
BB
BC
BD
BE
CA
CB
CC
CD
CE
1,1
1,3
1,5
1,7
1,9
3,1
3,3
3,5
3,7
3,9
5,1
5,3
5,5
5,7
5,9
X
1.00
2.00
3.00
4.00
5.00
2.00
3.00
4.00
5.00
6.00
3.00
4.00
5.00
6.00
7.00
Sample Scores
DA
DB
DC
DD
DE
EA
EB
EC
ED
EE
7,1
7,3
7,5
7,7
7,9
9,1
9,3
9,5
9,7
9,9
X
4.00
5.00
6.00
7.00
8.00
5.00
6.00
7.00
8.00
9.00
Summary statistics
(all samples, n=2)
S X = 125.00
N = 25
mu = 5.00
SS X = 100.00
sigma X 2 = 100/25=4.00
sigma X = 2.00
The standard error = the standard
deviation divided by the square
root of n, the sample size
• The formula works. And it works every
time.
Let’s see what sigmaX-bar can tell us
• We know that the mean of SAT/GRE scores = 500
and sigma = 100
• So 68.26% of individuals will score between 400 and
600 and 95.44% will score between 300 and 700
• REMEMBER THAT SAMPLE MEANS FALL
CLOSER TO MU, ON THE AVERAGE, THAN DO
INDIVIDUAL SCORES.
What happens when we take
random samples of SAT scores
with n=4?
• The standard error of the mean is sigma divided by the
square root of the sample size = 100/2=50.
• SAMPLE MEANS FALL INTO A NORMAL CURVE
EVEN BETTER THAN INDIVIDUAL SCORES
• 68.26% of the sample means (n=4) will be within 1.00
standard error of the mean from mu and 95.44% will be
within 2.00 standard errors of the mean from mu
• So, 68.26% of the sample means (n=4) will be between
450 and 550 and 95.44% will fall between 400 and 600
Let’s make the samples larger
• Take random samples of SAT scores, with 400 people in each
sample, the standard error of the mean is sigma divided by the
square root of 400 = 100/20=5.00
• 68.26% of the sample means will be within 1.00 standard
error of the mean from mu and 95.44% will be within 2.00
standard errors of the mean from mu.
• So, 68.26% of the sample means (n=400) will be between 495
and 505 and 95.44% will fall between 490 and 510.
• See how sample means get closer and closer to mu as sample
size increases!
The Central Limit Theorem
What happens as n increases?
• The sample means get closer to each other and to mu.
• Their average squared distance from mu equals the
variance divided by the size of the sample.
• Therefore, their average unsquared distance from mu
(which is called the standard error of the mean)
equals the standard deviation divided by the square
root of the size of the sample.
• The sample means fall into a more and more perfect
normal curve.
• These facts are called “The Central Limit
Theorem” and can be proven mathematically.
CONFIDENCE INTERVALS
We want to define two intervals
around mu:
One interval into which 95% of
the sample means will fall.
Another interval into which 99%
of the sample means will fall.
95% of sample means will fall in a symmetrical
interval around mu that goes from 1.960 standard
errors below mu to 1.960 standard errors above mu
• A way to write that fact in statistical language is:
CI.95: mu + 1.960 sigmaX-bar
or
CI.95: mu - 1.960 sigmaX-bar < X-bar < mu + 1.960 sigmaX-bar
As I said, 95% of sample means will fall in a
symmetrical interval around mu that goes from 1.960
standard errors below mu to 1.960 standard errors
above mu
• Take samples of SAT/GRE scores (n=400)
• Standard error of the mean is sigma divided by the square root
of n=100/ 400 = 100/20.00=5.00
• 1.960 standard errors of the mean with such samples = 1.960
(5.00)= 9.80
• So 95% of the sample means with n=400 can be expected to fall
in the interval 500+9.80
• 500-9.80 = 490.20 and 500+9.80 =509.80
CI.95: mu + 1.960 sigmaX-bar = 500+9.80
CI.95: 490.20 < X-bar < 509.20
or
99% of sample means will fall within 2.576
standard errors from mu
• Take the same samples of SAT/GRE scores (n=400)
• The standard error of the mean is sigma divided by the square
root of n=100/20.00=5.00
• 2.576 standard errors of the mean with such samples =
2.576 (5.00)= 12.88
• So 99% of the sample means can be expected to fall in the
interval 500+12.88
• 500-12.88 = 487.12 and 500+12.88 =512.88
CI.99: mu + 2.576 sigmaX-bar = 500+12.88
or
CI.99: 487.12 < the sample mean < 512.88
Notice that the 99% CI includes the
95% CI. With n=400, mu=500.00,
sigma=100.00
CI.95: 490.20 < X-bar < 509.20
CI.99: 487.12 < X-bar < 512.88
You do one
• Using SAT scores, with n=2500:
• Into what interval should 95% of the sample
means fall?
Interval around mu
• 95% of the sample means should fall within
1.960 standard errors of the mean from mu.
• Given that sigmaX-bar =2.00, you multiply
1.960 * sigmaX-bar = 1.960 x 2.00 = 3.92
Thus:
• 95% of the sample means should fall in an
interval that goes 3.92 points in both
directions around mu
• 500 – 3.92 = 496.08
• 500 + 3.92 = 503.92
• So 95% of sample means (n=2500) should
fall between 496.08 and 503.92
• CI.95: 496.08 < X-bar < 503.92