more confusion on cessna performance chart

A little while ago I started a thread about why a Cessna landing
peformance chart showed a climb wt limit that was pressure altitude
rather than density altitude dependant. Well now I am similarly
confused by a different performance table ( not a chart this time)
which shows the take off distance required for a C172N. This table
shows the distance required as a function of pressure altitude form 0
to 8000 ft ( in steps of 1000 ft) and temperature of 0 to 40 deg C in
steps of 10 deg C. This table comes from the flight manual of the
aircraft.

Now I thought it would be a useful addition to my Excel flight fomulas
to convert all this data into a graph of distance required vs density
altitude and fit an equation to it, so then I could just enter density
altitude and the program would calculate distance required ( with
corrections for wind etc)
Well the problem is that where the density altitude ranges overlap for
the different temperatures the distance required differs with the
lower the temp, the higher the distance required for the same density
altitude.

At around 8000 ft density altitude the difference was like 300 ft
between 0 and 40 deg C which is quite significant. I am pretty sure I
know how to calculate density altitude, but just in case here is how I
do it,

Take the pressure altitude and correct for temperature as follows.
eg for 3000 ft pressure altitude and 30 deg C. ISA temp would be
(15-3x2)=+9 C, so we are 21 C over ISA temp. 21*120 =2520 +3000
=5520 ft density altitude

So why would the takeoff distance required vary with temperature at
the same density altitude?, it goes against everything I understood
about peformance being a function of the air density.

Any help appreciated.
Terry
PPL downunder