Range Median

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Last modified: March 26, 2020

The Range Median function is used to calculate a median from a series of data giving a count within a range.  For the purposes of this explanation Median Household income will be used and the example data will be AGS 2004 data for Boulder County, Colorado.

The data as provided for AGS for Boulder looks like:

Var

Desc

Households

Running Total

HIXCYLT05

Income    $0 to $5,000

4233

4233

HIXCY0510

Income   $5,000 to $9,999

2991

7224

HIXCY1015

Income  $10,000 to $14,999

4277

11501

HIXCY1520

Income  $15,000 to $19,999

4364

15865

HIXCY2025

Income  $20,000 to $24,999

4756

20621

HIXCY2530

Income  $25,000 to $29,999

4991

25612

HIXCY3035

Income  $30,000 to $34,999

5483

31095

HIXCY3540

Income  $35,000 to $39,999

5109

36204

HIXCY4045

Income  $40,000 to $44,999

5180

41384

HIXCY4550

Income  $45,000 to $49,999

4709

46093

HIXCY5055

Income  $50,000 to $54,999

4784

50877

HIXCY5560

Income  $55,000 to $59,999

4701

55578

HIXCY6065

Income  $60,000 to $64,999

4720

60298

HIXCY6570

Income  $65,000 to $69,999

3637

63935

HIXCY7075

Income  $70,000 to $74,999

3943

67878

HIXCY7580

Income  $75,000 to $79,999

4110

71988

HIXCY8085

Income  $80,000 to $84,999

3680

75668

HIXCY8590

Income  $85,000 to $89,999

3057

78725

HIXCY9095

Income  $90,000 to $94,999

2785

81510

HIXCY95100

Income  $95,000 to $99,999

2523

84033

HIXCY10025

Income $100,0000 to $124,999

11234

95267

HIXCY12550

Income $125,000 to $149,999

6186

101453

HIXCY15020

Income $150,000 to $199,999

5015

106468

HIXCY20050

Income $200,000 to $249,999

1969

108437

HIXCY25050

Income $250,000 to $499,999

2463

110900

HIXCYGT500

Income $500,000 Plus

1523

112423

This is giving counts of households in each income range.  How do we calculate a median from the ranges?  Going back to the definition of a median, "a value in an ordered set of values below and above which there is an equal number of values (from dictionary.com)", we first have to find the range where the median point must exist.  There are 112,423 households in the above table. The median HH income is the household where 56,211 households are above and below that one. Looking at the running total above, we can see that the median must be within the $60,000 to $64,999 range.

So now that we know for certainty that the Median Income for Boulder County is between $60K and $65K, we would like to know more specifically what it is.  In reality, we don't have any information about the households in that range.  All 4720 households could be making $60,001 or they could all be making $64,999, so the median really could be anywhere in that range, but we can presume that most areas fit a bell curve and there is a fairly even distribution of households in that range.  We know that the median is 633 households into the range (56211-55578=633) or 13.4 % (100*633/4720=633).  So simplistically we can assume that the median is 13.4% of the way into the range or 670 HHs into the range (.134*5000+60000=60670).

Boulder County Household Income Chart

Now, just because we don't want to be too simple and we want to produce the most accurate result possible, we can take our estimate one step further.  If we look at the above chart of HH incomes in Boulder, we see that there is a major drop off from the $60K-65K group to the $65K-70K group.  Above we presumed that there were an equal # of people making incomes within the entire $60-65K range, but we can assume that there is a falloff within the range the is similar to the falloff between the ranges and that HHs are more concentrated at the low end of the range.  So the RangeMedian function makes a very simplistic model of the curve (it only looks at 1 range above and 1 range below the median range.)  Using a little area under the curve calculus, we can more accurately place the median at 60653.  The exact details of the math are a little too complicated to put here, but the result is always very similar to the above simplistic method, but slightly more accurate.

RangeMedian Formula for calculating Median HH income

RANGEMEDIAN(0,[HIXCYLT05],5000,[HIXCY0510],10000,[HIXCY1015],15000,[HIXCY1520],20000,[HIXCY2025],25000,[HIXCY2530],30000,[HIXCY3035],35000,[HIXCY3540],40000,[HIXCY4045],45000,[HIXCY4550],50000,[HIXCY5055],55000,[HIXCY5560],60000,[HIXCY6065],65000,[HIXCY6570],70000,[HIXCY7075],75000,[HIXCY7580],80000,[HIXCY8085],85000,[HIXCY8590],90000,[HIXCY9095],95000,[HIXCY95100],100000,[HIXCY10025],125000,[HIXCY12550],150000,[HIXCY15020],200000,[HIXCY20050],250000,[HIXCY25050],500000,[HIXCYGT500])

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