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

Example:

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 library’s expenditure peer 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:

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
              5.4

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 standard scores from step 3—one for each per capita rate included in the LJ Index—are summed.

Example for a given library:

 .96                (standard score for visits per capita)
+ 1.57            (standard score for circulation per capita)
+ 1.23            (standard score for circulation of electronic materials per capita)
+ 1.10            (standard score for total program attendance per capita)
+ 1.51            (standard score for public internet terminal uses per capita)
+ 2.46            (standard score for Wi-Fi sessions per capita)
+ 2.73            (standard score for electronic retrievals per capita)
= 11.56          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 all libraries rated, preliminary scores from step 4 may include libraries with negative scores—for instance, -7.8. Therefore, a correction factor—8 for this example—is added to all preliminary scores so that no LJ Index scores are negative.

Step 6. To address outliers, libraries whose data, including their raw LJ Index scores, do not meet the 2 outlier criteria described in FAQ #20 are deleted at this point. They do not receive LJ Index scores and their data are not reported.

Step 7. The final LJ Index score is calculated by multiplying sum of the preliminary score from step 5 and the correction factor by 100 and then rounding to whole numbers.

Example for a given library:

11.56 + 8.0   =   19.56    (Preliminary score plus correction factor)

19.56 x 100  =   1956     (Final LJ Index score)