This is one of those lessons that I thought was Maths teaching folklore - but after a Complete Maths Teaching Together on Statistics, it appears it isn't.
A quick Google search didn't bring up any results so here's my best attempt at explaining.
I've been using this for all of my 25 years of Maths teaching and I'm fairly sure I "magpied" the idea from another teacher.
Mean from grouped frequency.
You will need:
2-3 small pieces of paper per student
10 envelopes (DL or A5 work best)
Blu tac
Students will need:
2-3 small pieces of paper
A pen
A calculator
The basics...
Write your class intervals on the front of the envelopes in "board handwriting".
Hand out slips of paper to students (2 -3 each depending on class size)
Ask students to write a number on each piece of paper within the range of your class intervals. Encourage non-integers.
Collect all students numbers in and place on top of the correct envelope. (I usually get students to do this themselves - if you can cope with a few minutes bedlam)
Ask selected to students to count how many pieces of paper are going into each envelope. They should write this number on the front of the envelope and seal the envelope.
Blu tac the envelopes to the board as the first column in your grouped frequency table.
Add the frequency column to the table.
Ask students how we could estimate the mean of all the classes numbers without opening the envelopes.
Proceed through the mean from a grouped frequency calculation.
Hand back out envelopes to students and ask them to total the numbers in each envelope.
Complete actual mean calculation.
So that makes perfect sense to me - shout if you need some photos to explain it better.