Difference between revisions of "Statistical formulas documentation"
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<u>'''When the two populations have equal variances'''</u> | <u>'''When the two populations have equal variances'''</u> | ||
| − | <math>t= | + | <math>t = \frac{m_1-m_2}{/sqrt{(/frac{(n_1-1)(s_1)^2+(n_2-1)(s_2)^2}{n_1+n_2-2)})(\frac{1}{n_1}+\frac{1}{n_2})}}</math> |
Revision as of 16:42, 6 August 2013
mTAB Median Calculation from Income Brackets
| Approximate Annual HH Income | |
| Median | 71,180 |
| Unweighted Sample Total Count | 10,811 |
| Approximate Annual HH Income | Weighted Response (1) Accumulated Response |
||
| Less than $15,000 || width="150"|11,714 || 11,714 |- | $15,000 - $24,999 || 46,054 || 57, 768 |- | $25,000 - $34,999 || 83,965 || 141,733 |- | $35,000 - $44,999 || 102,093 || 243,826 |- | $45,000 - $59,999 || 155,721 || '''399,546''' |- | $60,000 - $74,999 || 161,435 || '''560,981''' <--''Median will fall here'' (3) |- | $75,000 - $99,999 || 193,540 || 754,521 |- | $100,000 - $124,999 || 134,706 || 889,227 |- | $125,000 - $149,999 || 59,748 || 948,975 |- | $150,000 - $199,999 || 41,971 || 990,946 |- | $200,000 - $249,999 || 16,391 || 1,007,337 |- | $250,000 or More | 32,409 | 1,039,746 | |
| Weighted Subset Total Count | 1,039,746 | ||
| Weighted Sample Total Count | 1,255,411 | ||
| (1) | Calculated Accumulated Weighted Response | |||||
| (2) | Divide total (1,039,746) by 2=519,873 | 519,873 | ||||
| (3) | Find first value in Accumulated Response column that is greater than step 2 value | |||||
| The median will fall between the $60,000-$74,999 bracket | ||||||
| (4) | Step 2 amount (519,873) MINUS preceding break accumulated response 399,546 = | 120,327 | ||||
| (5) | Acc. Response where Meidan will fall 560,981 MINUS preceding break 399,546 = | 161,435 | ||||
| (6) | Step 4 Divided by Step 5 | 0.74536 | ||||
| (7) | Multiply Step 6 by the range 14,999 ($60,000-$75,999) | 11180 | ||||
| (8) | Add Step 7 to bottom of range $60,000 || '''71,180''' |} '''mTAB Mean/Weighted Average Calculation from Income Brackets {|width=275px |- |Approximate Annual HH Income |- | Mean/Weighted Average || 83,610 |- | Unweighted Sample Total Count || 10,811 |} {| |- | || (A) || || || (B) || (C) |- | width="225"|Approximate Annual HH Income || || STAT1 || STAT2 || Midpoint |- | Less than $15,000 | 11,714 | 1 | 14,999 | 7,500 | 87,857,249 |
| $15,000 - $24,999 | 46,054 | 15,000 | 24,999 | 20,000 | 921,059,004 | |
| $25,000 - $34,999 | 83,965 | 25,000 | 34,999 | 30,000 | 2,518,899,346 | |
| $35,000 - $44,999 | 102,093 | 35,000 | 44,999 | 40,000 | 4,083,654,266 | |
| $45,000 - $59,999 | 155,721 | 45,000 | 59,999 | 52,500 | 8,175,254,132 | |
| $60,000 - $74,999 | 161,435 | 60,000 | 74,999 | 67,500 | 10,896,752,251 | |
| $75,000 - $99,999 | 193,540 | 75,000 | 99,999 | 87,500 | 16,934,669,636 | |
| $100,000 - $124,999 | 134,706 | 100,000 | 124,999 | 112,500 | 15,154,345,342 | |
| $125,000 - $149,999 | 59,748 | 125,000 | 149,999 | 137,500 | 8,125,258,359 | |
| $150,000 - $199,999 | 41,971 | 150,000 | 199,999 | 175,000 | 7,344,910,850 | |
| $200,000 - $249,999 | 16,391 | 200,000 | 249,000 | 225,000 | 3,688,025,252 | |
| $250,000 or More | 32,409 | 250,000 | 300,000 | 275,000 | 8,912,452,979 | |
| Weighted Subset Total Count | 1,039,746 | 86,933,138,666 | ||||
| Weighted Sample Total Count | 1,255,411 |
- (1) Find Midpoint of data ranges - Column (B)
- (2) Multiply Weighted Counts (A) by Midpoints (B) to generate (C)
- (3) Divide the sum of column (C) by the total weighted response at the bottom of column (A)...
- 86,933,138,666 divided by 1,039,746 = 83,610
- You will notice the calculated average matches the mTAB produced average
Standard Deviation Calculation
For categorized questions, each response is assigned 1 or 2 stat weights. If a single weight is assigned, then this is the value used to calculate the standard deviation. If 2 weights are provided, the midpoint is used.
D = Question Mean - Stat Value as described above
SS = Sum of Squares, D*D*Weighted Response Count, for all table responses
Sample = Sum of all Weighted Response Counts for all table responses
Standard Deviation = SQRT(SS/Sample-1));
The calculation is the same for continuous variables except the actual data values are used instead of stat weight.
D = Question Mean - Response Value
SS = Sum of Squares, D*D*Respondent Weight Count for each response
Sample = Sum of all Respondent Weights for each response
Standard Deviation = SQRT(SS/Sample-1));
Comparison of two population means using T-Statistic
When the two populations have equal variances
\(t = \frac{m_1-m_2}{/sqrt{(/frac{(n_1-1)(s_1)^2+(n_2-1)(s_2)^2}{n_1+n_2-2)})(\frac{1}{n_1}+\frac{1}{n_2})}}\)
