How Does Sample Size Affect Standard Deviation

Chapter 9. Inferential statistics: from samples to populations

1. What is the relationship betwixt sampling variability and standard errors?

Standard errors are measures of sampling variability.

ii. Assume is 2.twoscore and the sample size is 36. What will go if you modify the sample size to:

a. 72

In that location are two means to do this.

one.) Solve for due south:

is ii.twoscore and the sample size is 36, and since is defined as and estimated equally , the standard difference must exist:

Now plug the standard deviation into the equation and get the new standard error:

b. ix

The new sample size is one fourth as big, and then: and the new standard error volition exist twice as big every bit the original one:

c. 144

The new sample is iv times as large, so: and the new standard error is half every bit large every bit the original one:

3. Assume is 3.60 and your estimate for is 9.00. Assuming your sample size does not alter, what will be if you could change to:

a. 12.0

Changing from 9.0 to 12.0 will increase the standard fault of the mean by 12/9 = 1.33, which will requite you iv.eight instead of three.6.

b. 4.five

Changing from 9.0 to 4.v volition decrease the standard error of the mean by 4.five/9 = 0.5, which volition give you 1.viii instead of 3.vi.

c. xiii.5

Changing from 9.0 to thirteen.v volition increase the standard error of the mean by 13.5/9 = 1.5, which will give you v.4 instead of 3.vi.

iv. If the sample's standard deviation tells you lot how adept the sample'due south hateful is as a description of the typical person in the sample, the standard error of the mean tell yous how good the sample'southward mean is equally a description of what? In other words, if the sample's standard deviation tells y'all how far the sample'southward mean is from the typical person in the sample, the standard error of the mean tells you how far the sample's mean is likely to exist from what?

How far from the population's mean.

v. Summate for the post-obit xi samples:

	n	s
a.	36	6.0	ane.0000
b.	36	8.0	ane.3333
c.	36	12.0	2.0000
d.	49	6.0	0.8571
due east.	49	8.0	ane.1428
f.	72	12.0	1.4142
g.	98	half-dozen.0	0.6061
h.	98	8.0	0.8081
i.	98	12.0	1.2122
j.	144	12.0	i.0000
l.	144	eight.0	0.6667

6. Examine the answers you obtained for question 5.

a. What upshot does doubling the sample size take on when southward doesn't change?

Information technology is an inverse square relation. Multiplying the sample size by two divides the standard error by the square root of ii. The new will be:

b. What event does quadrupling the sample size accept on when s doesn't change?

Multiplying the sample size past 4 divides the standard error by the square root of 4. The new will be . Information technology will be half every bit large as the original.

c. What upshot does doubling s accept on when the sample size doesn't change?

The standard error of the mean is directly proportional to the standard deviation. Doubling s doubles the size of the standard mistake of the hateful.

east. What effect does increasing s have on when the sample size doesn't alter?

Increasing southward increases the size of the standard error of the mean past the same cistron.

7. Overall, what is the relation between sample size and ?

Bigger samples produce smaller standard errors. The relation is an inverse square root relation: increasing the sample size past a factor of C decreases the standard error by a gene of one over the square root of C.

eight. Summate for the following six pairs of samples:

	sample 1		sample 2
	northward	south	n	s
a.	45	5.50	45	five.l	1.1595
b.	60	5.50	60	v.50	one.0042
c.	60	8.l	threescore	eight.l	1.5519
d.	45	5.50	60	5.50	1.0846
due east.	45	11.00	45	eleven.00	2.3190
f.	180	5.50	180	5.50	0.5798

9. Examine the answers for question 8.

a. What effect does increasing s accept on when the sample size doesn't change?

When s increases, increases.

b. What consequence does increasing the sample size have on when s doesn't change?

When n increases, decreases.

c. What effect does increasing the size of i sample accept on when southward and the other sample size don't change?

When one sample size increases, decreases.

d. What event does doubling the sample size take on when s doesn't alter?

The new will become the onetime × 0.707107

eastward.What effect does quadrupling the sample size have on when due south doesn't change?

The new volition go the old × 0.5

f. Overall, what is the relation between sample size and ?

Bigger samples smaller standard errors. The relation is an inverse square root relation: increasing the sample size by a factor of C decreases the standard mistake by a gene of i over the square root of C.