User Formulas and Guidelines in Performing Empirical Research and Surveys

User Formulas and Guidelines in Performing Empirical Research and Surveys

(under construction, last updated 10-29-09)

Calculating Samp= ling Error:

1. General formula:

   &n= bsp;

   Where p =3D percentage in decimal form, e.g. .56 <= /p>
        =        N =3D population size

        =        n =3D sample size

2. Formula for samples when the s= ample is small relative to the population (< 10% of the population):

Same as above but drop the correction factor on the right hand side of the equat= ion: (N - n) / (N -1)

3. Formula for when you do not kn= ow what p is, that is, when estimating the needed sample size before the surve= y is performed:

        =           =       Sampling Error =3D 1 / sq. rt. of n       wher= e n is sample size

        =             &nb= sp; You can also solve for n if you want to start with a desired sampling error. In that case the formula is:

        =             &nb= sp;        n =3D 1 / SE squared

Random Number Table (f= or choosing a random sample when you have a numbered list or for generating ra= ndom telephone numbers):

TABLE OF RANDOM NUMBERS=

_{3=
9634 62349
74088 65564 16379 19713 39153 69459 17986 24537}

_{1=
4595 35050
40469 27478 44526 67331 93365 54526 22356 93208}

_{3=
0734 71571
83722 79712 25775 65178 07763 82928 31131 30196}

_{6=
4628 89126
91254 24090 25752 03091 39411 73146 06089 15630}

_{4=
2831 95113
43511 42082 15140 34733 68076 18292 69486 80468}

_{8=
0583 70361
41047 26792 78466 03395 17635 09697 82447 31405}

_{0=
0209 90404
99457 72570 42194 49043 24330 14939 09865 45906}

_{0=
5409 20830
01911 60767 55248 79253 12317 84120 77772 50103}

_{9=
5836 22530
91785 80210 34361 52228 33869 94332 83868 61672}

_{6=
5358 70469
87149 89509 72176 18103 55169 79954 72002 20582}

_{7=
2249 04037
36192 40221 14918 53437 60571 40995 55006 10694}

_{4=
1692 40581
93050 48734 34652 41577 04631 49184 39295 81776}

_{6=
1885 50796
96822 82002 07973 52925 75467 86013 98072 91942}

_{4=
8917 48129
48624 48248 91465 54898 61220 18721 67387 66575}

_{8=
8378 84299
12193 03785 49314 39761 99132 28775 45276 91816}

_{7=
7800 25734
09801 92087 02955 12872 89848 48579 06028 13827}

_{2=
4028 03405
01178 06316 81916 40170 53665 87202 88638 47121}

_{8=
6558 84750
43994 01760 96205 27937 45416 71964 52261 30781}

_{7=
8545 49201
05329 14182 10971 90472 44682 39304 19819 55799}

_{1=
4969 64623
82780 35686 30941 14622 04126 25498 95452 63937}

_{5=
8697 31973
06303 94202 62287 56164 79157 98375 24558 99241}

_{3=
8449 46438
91579 01907 72146 05764 22400 94490 49833 09258}

_{6=
2134 87244
73348 80114 78490 64735 31010 66975 28652 36166}

_{7=
2749 13347
65030 26128 49067 27904 49953 74674 94617 13317}

_{8=
1638 36566
42709 33717 59943 12027 46547 61303 46699 76243}

_{4=
6574 79670
10342 89543 75030 23428 29541 32501 89422 87474}

_{1=
1873 57196
32209 67663 07990 12288 59245 83638 23642 61715}

_{1=
3862 72778
09949 23096 01791 19472 14634 31690 36602 62943}

_{0=
8312 27886
82321 28666 72998 22514 51054 22940 31842 54245}

_{1=
1071 44430
94664 91294 35163 05494 32882 23904 41340 61185}

_{8=
2509 11842
86963 50307 07510 32545 90717 46856 86079 13769}

_{0=
7426 67341
80314 58910 93948 85738 69444 09370 58194 28207}

_{5=
7696 25592
91221 95386 15857 84645 89659 80535 93233 82798}

_{0=
8074 89810
48521 90740 02687 83117 74920 25954 99629 78978}

_{2=
0128 53721
01518 40699 20849 04710 38989 91322 56057 58573}

_{0=
0190 27157
83208 79446 92987 61357 38752 55424 94518 45205}

_{2=
3798 55425
32454 34611 39605 39981 74691 40836 30812 38563}

_{8=
5306 57995
68222 39055 43890 36956 84861 63624 04961 55439}

_{9=
9719 36036
74274 53901 34643 06157 89500 57514 93977 42403}

_{9=
5970 81452
48873 00784 58347 40269 11880 43395 28249 38743}

_{5=
6651 91460
92462 98566 72062 18556 55052 47614 80044 60015}

_{7=
1499 80220
35750 67337 47556 55272 55249 79100 34014 17037}

_{6=
6660 78443
47545 70736 65419 77489 70831 73237 14970 23129}

_{3=
5483 84563
79956 88618 54619 24853 59783 47537 88822 47227}

_{0=
9262 25041
57862 19203 86103 02800 23198 70639 43757 52064}

Statistical Proc= edures to Use When Testing a Bivariate Relationship:

     The proper procedure to use is determined by th= e levels of measurement in the independent and dependent variable. The table below s= hows the procedure most usually performed.

            &nb= sp;            =             &nb= sp;            =             &nb= sp;        Independent Variable <= /span>

Dependent

Nominal

Ordinal

Ratio

Nominal

X-tab

X-tab

Collapse I.V. & X-tab<= span style=3D'color:black'>

Ordinal

X-tab

X-tab

Collapse I.V. & X-tab<= span style=3D'color:black'>

Ratio

X-tab

or ANOVA

X-tab

or ANOVA

Collapse both variables & X-T= ab

or scatterpl= ot/regression

     The Crosstabulation pr= ocedure (X-tab) works best when you have no more than three rows in the table. That means no more than three values for the dependent variable. Of course, you = can always collapse rows to get this, or you can compare measures of central tendency (medians or means). =

  = ;   It really does not matter how many columns you have, but you had better have few enough so that you have a sufficient number of cases in each cell. You = can collapse columns to get this.

How to Read and Explain a Crosstabulation:

The f= ormula for reading any row in a crosstabulation is as follows. This wording works for almost any crosstabula= tion. :

"As the independ= ent variable changes from ___ to ___ (reading left to right), the percentage of those who are _______ (whatever that row is in the dependent variable) shif= ts (increases, decreases or however the percentages change, maybe even up and down) from ___ to ___. "

Chi Square (for = hand-calculating significance of a crosstabulation table):<= /o:p>

Where Fe is frequency expected by chance if there were no relationship (use formu= la below)

And Fo is the frequency = observed in your actual sample in each cell

Significance Tab= le for Chi Square:

After you calculate chi square for a table, you must interpret whether it is big enough to indicate that any relationship in the table was significant. You = use the following formula to see which row to use in the significance table bel= ow:

degrees of freedom =3D (# rows-1)(# columns-1)

Degrees of freedom

Min chi sq for a sig level of .10 or 10%=

Min chi sq for a sig level of .05 or 5%<= /o:p>

Min chi sq for a sig level of .01 or 1%<= /o:p>

1

1.64

2.71

5.41

2

3.22

4.60

7.82

3

4.64

6.25

9.84

4

5.99

7.78

11.67

5

7.29

9.24

13.39

6

8.56

10.64

15.03

7

9.80

12.02

16.62

8

11.03

13.36

18.17

Examples: Suppose you have a crosst= abulation with 2 rows and 3 columns. You use the formulas and compute the chi square = and find it to be 4.25. The degrees of freedom are (2-1)(3-1) =3D (1)(2) =3D 2, so you will use the second row of the table above. A value of 4.25 lies between the .10 column and the .05 colum= n. So the significance level is somewhere between .10 and .05, or between 10 p= ercent and 5 percent. Because in social science we insist that the significance level be at 5% or lower, we cannot conclude that what we saw in this particular crosstabulation = is statistically significant. = The proper conclusion to draw could be worded as follows. This is not a significant relationship because more than a 5 percent chance exists that this relationship could be found in a sample when no relationship exists in the population.

Had the value of the chi square been greater than 4.60, say 5.33, we would have concluded: This is a significant relationship because less than a 5 percent chance exists that this relation= ship could be found in a sample when no relationship exists in the population.

To take the example to another possibility, suppose that the chi square turns = out to be really large, say 64.32. That is entirely possible when the sample is relatively large and the percentage shifts in the cros= stabulation rows are also large. We would then conclude: This is a significant relationship because less than a 1 percent ch= ance exists that this relationship could be found in a sample when no relationsh= ip exists in the population.

Measures of Strength of Associ= ation

     Which statistic you use depends on levels of measurement of the two variables. Wh= ile many statistics are available, here are the easiest ones to use among those prov= ided by MicroCase.

Two nominal measures or one ordinal= : use Cramer’s V. Tha values range from 0 to 1. The c= loser to 1, the stronger the relationship.

Two ordinal measures: Kendall’s Tau B if table is square and Kendal= l’s Tau C if rectangular. The values range from -1 through 0 to +1. The closer to 1, the stronger the relationship.

Two interval or ratio: use Pearson correlation. The values range from -1 through 0 to +1. The closer = to 1, the stronger the relationship.

Interpreting Str= ength of Measures of Association

     Here are some guidelines on the adjectives you use for the varying values of statistical measures of association. You ONLY use these descriptors if the relationship= was statistically significant. No descriptors should be used for insignificant relationships, no matter what the value of any statistical measure of association. (Guidelines suggested by Le Roy, Research Methods in Political Science. Bo= ston: Wadsworth, 2009.)

For Cramer’s V and Tau B an= d C:

Less than + or - 0.10: very weak

+ or -0.10 to 0.19: weak=

+ or - 0.20 to 0.29: moderate

+ or - 0.30 or above: strong

For Correlations:

Less than + or - 0.25: extremely weak

+ or -0.25 to 0.34: weak=

+ or - 0.35 to 0.39: moderate

+ or - 0.40 or larger: strong

     For example, sup= pose we had a Crosstab between two nominal level variables and the Cramer’= s V was 0.23 while the chi square was 1.75 with p =3D 0.11. In this case, the relationship is statistically insignificant, so we do not need to interpret= the Cramer’s V. At most, we might conclude that the sample was too small = to draw any conclusions about the strength of the relationship and then call f= or further study with a larger sample. You should know this because you should know that the chi square is very sensitive to sample size.

     But suppose the = chi square was larger so that p =3D 0.021. Now it is a significant relationship= and the Cramer’s V of 0.23 tells us that it is a “moderately strong= ” relationship.

Dependent	Nominal	Ordinal	Ratio
Nominal	X-tab	X-tab	Collapse I.V. & X-tab<= span style=3D'color:black'>
Ordinal	X-tab	X-tab	Collapse I.V. & X-tab<= span style=3D'color:black'>
Ratio	X-tab or ANOVA	X-tab or ANOVA	Collapse both variables & X-T= ab or scatterpl= ot/regression

Degrees of freedom	Min chi sq for a sig level of .10 or 10%=	Min chi sq for a sig level of .05 or 5%<= /o:p>	Min chi sq for a sig level of .01 or 1%<= /o:p>
1	1.64	2.71	5.41
2	3.22	4.60	7.82
3	4.64	6.25	9.84
4	5.99	7.78	11.67
5	7.29	9.24	13.39
6	8.56	10.64	15.03
7	9.80	12.02	16.62
8	11.03	13.36	18.17