Social Research Methods: Qualitative and Quantitative Approaches

QUALITATIVE AND QUANTITATIVE SAMPLING

Let us say you draw a sample of nine 12-year-old
children. You weigh them and find that their average
weight, the mean, is 90 pounds with a standard devi-
ation of 36 pounds. You want to create a confidence
interval around your best estimate of the population
parameter (the mean weight for the population of all
12-year-olds). You symbolize the population param-
eter with the Greek letter μ.
Here is how to figure out a confidence interval for
the population mean based on a simple random
sample. You estimate a confidence level around μ by
adding and subtracting a range above and below the
sample mean, your best estimate of μ.
To calculate the confidence interval around the
sample mean, you first calculate something called
thestandard error of the mean. Call it standard error
for short. It is your estimate of variability in the
sampling distribution. You use another Greek letter,

σ, to symbolize the standard deviation and add the

letter mas a subscript to it, indicating that it is your

estimate of the standard deviation in the sampling

distribution. Thus, the standard error comes from the

standard deviation in the sampling distribution of all

possible random samples from the population.

You estimate the standard deviation of the

sampling distribution by getting the standard devia-

tion of your sample and adjusting it slightly. To sim-

plify this example, you skip the adjustment and

assume that it equals the sample standard deviation.

To get the standard error, you adjust it for your sample

size symbolized by the letter N. The formula for it is:

Let us make the example more concrete. For the

example, let us look at weight among nine 12-year-

olds. For the sampling distribution of the mean you

use a mean of 90 pounds and a standard deviation

of 36/3 12 (note the square root of 9 3). The con-

fidence interval has a low and upper limit. Here are

formulas for them.

Lower limit M Z.95σm

Upper limit M Z.95σm

In addition to the σmthere are two other symbols

now:

M in the formula stands for mean in your sample.

Z.95stands for the z-score under a bell-shaped or

normal curve at a 95 percent level of confidence (the

most typical level). The z-score for a normal curve is

a standard number (i.e., it is always the same for 95

percent level of confidence, and it happens to be

1.96). We could pick some confidence level other

than 95 percent, but it is the most typical one used.

You now have everything you need to calculate

upper and lower limits of the confidence interval. You

compute them by adding and subtracting 1.96 stan-

dard deviations to/from the mean of 90 as follows:

Lower limit 90 (1.96)(12) 66.48

Upper limit 90 (1.96)(12) 113.52

sm=

s

2 N

EXPANSION BOX 2

Confidence Intervals

The confidence interval is a simple and very powerful
idea; it has excellent mathematics behind it and some
nice formulas. If you have a good mathematics back-
ground, this concept could be helpful. If you are ner-
vous about complex mathematical formulas with
many Greek symbols, here is a simple example with
a simple formula (a minimum of Greek). The interval
is a range that goes above and below an estimate of
some characteristic of the population (i.e., population
parameters), such as its average or statistical mean.
The interval has an upper and lower limit. The
example illustrates a simplified way to calculate a
confidence interval and shows how sample size and
sample homogeneity affect it.

Lower Limit
of Interval

Upper Limit

of Interval

SAMPLE

MEAN

Confidence Interval

(continued)

N