April 19, 2018

The LJ Index: Score Calculation Algorithm

Score calculation algorithm

LJ Index of Public Library Service scores are calculated as described in the steps below. Libraries must meet four inclusion criteria to receive an LJ Index score.

Step 1. For each library, per capita rates for the four statistical indicators described above are calculated. A per capita rate is defined as:

statistical indicator
————————————— = statistical indicator per capita rate
service area population


205,150 visits annually
———————————————  = 11.4 visits per capita
17,990 service area population

Step 2. For each of the of nine expenditure groups, mean (average) per capita rates for the four indicators are calculated.

Next, for each of the expenditure groups, the standard deviations of the four per capita indicators are calculated. A standard deviation is a statistical measurement of variation within a group of data—how spread out the data are.

Example results for the $200K-$399K peer (expenditure) group :

Mean visits per capita:    6.2

Standard deviation of visits per capita: 5.4

Step 3. A standard score (a statistical measurement known also as a “Z-score”) is calculated for each of a given library’s four per capita rates. Calculations take this form:

per capita statistic for given library  –  mean per capita rate for library’s peer group
standard deviation of per capita statistic for library’s peer group

Example for the $200K-$399K peer group:

11.4 visits per capita for this library 

 6.2 mean visits per capita for peer group

 5.4 standard deviation of visits per capita for peer group

       11.4 –  6.2
—————————  =  .96 standard score for library’s visit per capita statistic

Note: Since the library’s visits per capita rate in this example is higher than the mean visits per capita for the peer group, the library’s standard score is positive. Had the library’s visits per capita rate been less than the peer group mean of 6.2, then the library’s standard score for visits per capita would be negative (less than zero).

Step 4. The four standard scores from step 3—one for each per capita rate—are summed.

Example for a given library:

 .96            (standard score for visits per capita)
+ 1.57            (standard score for circulation per capita)
+ 1.10            (standard score for total program attendance per capita)
+ 1.51            (standard score for public Internet terminal uses per capita)
5.14            Library’s preliminary LJ Index score

Step 5. As noted in step 3, some libraries’ standard scores can be negative (less than zero), as can the preliminary LJ Index score. Among the entire 7,115 libraries rated, preliminary scores from step 4 included libraries scoring as low as -5.5. Therefore, a correction factor of 6 was added to all preliminary scores so that no LJ Index scores are negative.

Step 6. The final LJ Index score is calculated by multiplying the results from step 5 by 100 and then rounding to whole numbers.

Example for a given library:

5.14 +  6.0   =   11.14    (Preliminary score plus correction factor)

11.14 X 100 =   1114     (Final LJ Index score)