I am about to buy a new football, to play on grass with my children. They are not very advanced, but we play regularly and my older child is nearly 150cm tall.
How can I decide which size to get for them, such as between a size 4 and size 5?
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FIFA provides Youth Football Specification Recommendations that are based on research for youth academy football conducted in Europe, Mexico and Japan. This document can be downloaded here (note, PDF).
The size and weight recommendations are based on age, progressing from size 3/4L at under-9s to size 4/5L at under-13s, and size 5 for all over-13s age groups (L denoting "lightweight").
The apparent intent of the progression is allowing young players to have a size and weight suitable for their physical ability to control the ball or propel it a reasonable distance, with weight gradation providing opportunity to develop coordination by making the ball bigger but not heavier, then increasing the weight to adjust for increased power, until attaining "full size" ball capability.
Following a similar progression is likely to provide the best development, and is found on page 2 of the document linked above.
US Soccer has a handy chart:
Above the ages in this chart, use size 5.