Difference between revisions of "Statistical formulas documentation"
| Line 53: | Line 53: | ||
| <span style="color:red">(3)</span> || Find first value in Accumulated Response column that is greater than step 2 value | | <span style="color:red">(3)</span> || Find first value in Accumulated Response column that is greater than step 2 value | ||
|- | |- | ||
| − | | || The ''median will fall between the | + | | || The ''median will fall between the 60,000-74,999'' bracket |
|- | |- | ||
| <span style="color:red">(4)</span> || Step 2 amount (<span style="color:red">519,873</span>) MINUS preceding break accumulated response '''399,546''' = || 120,327 | | <span style="color:red">(4)</span> || Step 2 amount (<span style="color:red">519,873</span>) MINUS preceding break accumulated response '''399,546''' = || 120,327 | ||
| Line 61: | Line 61: | ||
| <span style="color:red">(6)</span> || Step 4 Divided by Step 5 || 0.74536 | | <span style="color:red">(6)</span> || Step 4 Divided by Step 5 || 0.74536 | ||
|- | |- | ||
| − | | <span style="color:red">(7)</span> || Multiply Step 6 by the range 14,999 ( | + | | <span style="color:red">(7)</span> || Multiply Step 6 by the range 14,999 (60,000-75,999) || 11180 |
|- | |- | ||
| − | | <span style="color:red">(8)</span> || Add Step 7 to bottom of range | + | | <span style="color:red">(8)</span> || Add Step 7 to bottom of range 60,000 || '''71,180''' |
|} | |} | ||
| Line 85: | Line 85: | ||
| width="225"|Approximate Annual HH Income || || STAT1 || STAT2 || Midpoint | | width="225"|Approximate Annual HH Income || || STAT1 || STAT2 || Midpoint | ||
|- | |- | ||
| − | | Less than | + | | Less than 15,000 || width="150"|11,714 || width="150"|1 || width="150"|14,999 || width="150"|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 Subset Total Count || '''1,039,746''' || || || || '''86,933,138,666''' | ||
Revision as of 17:48, 8 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 | 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 Median 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
| Approximate Annual HH Income | |
| Mean/Weighted Average | 83,610 |
| Unweighted Sample Total Count | 10,811 |
| (A) | (B) | (C) | |||
| 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
\(t = \dfrac{m_1-m_2}{\sqrt{\dfrac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}(\dfrac{1}{n_1}+\dfrac{1}{n_2})}}\)
Where
m1 = Mean of the 1st sample
s1 = Standard Deviation of the 1st sample
n1 = Un-weighted Sample of the 1st sample
m2 = Mean of the 2nd sample
s2 = Standard Deviation of the 2nd sample
n2 = Un-weighted Sample of the 2nd sample
Decision rules:
If |t|<1.65 then the two populations are NOT significantly different at 90%
If |t|≥1.65 then the two populations ARE significantly different at 90%
If |t|<1.95 then the two populations are NOT significantly different at 95%
If |t|≥1.95 then the two populations ARE significantly different at 95%
When the two populations have UNEQUAL variances
\(t = \dfrac{m_1-m_2}{\sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}}}\)
Where
m1 = Mean of the 1st sample
s1 = Standard Deviation of the 1st sample
n1 = Un-weighted Sample of the 1st sample
m2 = Mean of the 2nd sample
s2 = Standard Deviation of the 2nd sample
n2 = Un-weighted Sample of the 2nd sample
Decision rules:
If |t|<1.65 then the two populations are NOT significantly different at 90%
If |t|≥1.65 then the two populations ARE significantly different at 90%
If |t|<1.95 then the two populations are NOT significantly different at 95%
If |t|≥1.95 then the two populations ARE significantly different at 95%
