Large Amplitude Pendulums
Without the smallangle approximation the differential equation is
not the simple harmonic motion equation and the period is not
independent of the amplitude. The exact period can't be written in
simple closed form, but it can be written as an elliptic integral of
the first kind (which can be numerically approximated) or written as
a power series expansion. Short of that, various other
approximations are also available. Plot one or two of the
approximations you learn about below on the same graph as your
experimental data.

The Large Angle Pendulum Period by Millet, TPT, March 2003

A Simple Formula for the LargeAngle Pendulum Period by Kidd and Fogg, TPT, February 2002

An accurate formula for the period of a simple pendulum oscillating beyond the small angle regime by Lima and Arun, AJP, Oct 2006

LargeAngle Motion of A Simple Pendulum Physics 258/259, 2005

Approximations for the period of the simplependulum based on the arithmeticgeometric mean by Carvalhaes and Suppes, AJP, December 2008

Approximation expressions for the largeangle period of a simple
pendulum revisited by Amrani and Beaudin, Rev, Mex, Fis. June 2008

Comment on 'Approximation for a largeangle simple pendulum period' by Yuan and Ding, Eur. J. Phys., L79L82, 2009

Reply to 'Comment on "Approximation for a largeangle simple pendulum period"' by Belendez, Rodes, Belendez, Hernandez, Eur. J. Phys., L83L86, 2009

An approximate expression for the large angle period of a simple pendulum by Parwani, Eur. J. Phys, 3739, 2004

Series expression for exact period from Wikipedia Pendulum_(mathematics)

Infinite series for exact period from Wikipedia Pendulum

The Simple Pendulum Lab by James Parks at Univ Tenn

The Simple Pendulum Lab at Youngstown State Univ

Period Calculator at Hyperphysics

Physics Notes on Pendulum Period at MIT, May 2009