## What you need to know

### Estimating The Mean

When we’re dealing with information displayed in a grouped frequency table – Revise grouped frequency tables if you’re not sure what that is – it’s impossible for us to calculate the actual mean because we don’t know each of the individual values.

So, instead, we try our best to estimate the mean.

## How to Estimate the Mean

The method for estimating the mean can be outlined as follows.

1.Add a new column to the table containing the midpoints of each group.

2.Multiply each midpoint by the frequency of that group and add up the results.

3.Divide that value by the total frequency to get the estimate of the mean.

Remember, it is just an estimate. The actual mean could be noticeably different.

## Example 1: Estimating the Mean

Cindy asked a collection of people how many hours they spent playing video games in the last week. Her results are shown in the table below. Estimate the mean number of hours spent playing video games from Cindy’s sample.

As outlined above, we need to calculate the midpoint of each class, as shown below.

Since we’re assuming that all 12 people in the first group played for 2 hours, then we have 12 lots of 2 to add up.

Apply this logic to each group and we get

\text{estimated mean}=\dfrac{(12\times 2)+(18\times 6)+(19\times 10)+(6\times 14)+(2\times 18)}{12+18+19+6+2}

Putting this (carefully) into a calculator, we get our answer:

\text{estimated mean }=7.75\text{ hours}## Example 1: Estimating the Mean

The grouped frequency table below shows data on the weights of 117 cats. Calculate an estimate for the mean weight of the cats.

As before, we need to note down the mid points of each group.

We then multiply each midpoint by it’s frequency and add them all together and divide by the total frequency as shown below

\text{mean estimate} = \dfrac{(22\times 2.5)+(14\times3.25)+(39\times 3.75)+(12\times 4.25)+(13\times 5.25)}{117}= 3.7 \text{ kg (1dp)}

### Example Questions

The first thing we need to do is add an additional midpoint column to our table. Since we have a range of journey times, we have to assume that all journeys are exactly in the middle of the range. This is probably not the case in reality, which is why this is an estimated mean rather than the actual mean.

The first midpoint, the point which is half-way between 0 minutes and 10 minutes, is 5 minutes. If you are unsure how to calculate a midpoint, add up the two values and divide by 2:

\dfrac{0 + 10}{2}=5

Once you have calculated all the midpoints, you should have a table that looks like the table below:

We then need to multiply the frequency of each group by its midpoint. This will give us the estimated total journey time for the number of journeys completed within each range of journey time.

2\times5\text{ minutes}= 10\text{ minutes}

45\times15\text{ minutes}= 675\text{ minutes}

25\times25\text{ minutes}= 625\text{ minutes}

3\times35\text{ minutes}= 105\text{ minutes}

We then need to add up all these new values. This will give us the estimated total journey time for all of the journeys completed.

10\text{ minutes}+675\text{ minutes}+625\text{ minutes}+105\text{ minutes} = 1415\text{ minutes}

This means that the total estimated journey time was 1415 minutes. By adding up the frequency column, we know that 75 journeys were completed (2+45+25+3=75\text{ journeys}). Since there were 75 journeys with an estimated total time of 1415 minutes, then the estimated mean can be calculated as follows:

\dfrac{1415 \text{ minutes}}{75\text{ journeys}} = 19 \text{ minutes to the nearest minute}

The first thing we need to do is add an additional midpoint column to our table. Since we have a range of long jump distances, we are assuming that all jumps are exactly in the middle of the range. This is probably not the case in reality, which is why this is an estimated mean rather than the actual mean.

The first midpoint, the point which is half-way between 0 cm and 50 cm, is 25 cm. If you are unsure how to calculate a midpoint, add up the two values and divide by 2:

\dfrac{0 + 50}{2}=25

Once you have calculated all the midpoints, you should have a table that looks like the table below:

We then need to multiply the frequency of each group by its midpoint. This will give us the estimated total jump length for the number of jumps completed within each range of jump length.

2\times25\text{cm}= 50\text{cm}

18\times75\text{cm}= 1,350\text{cm}

56\times125\text{cm}= 7,000\text{cm}

32\times175\text{cm}= 5,600\text{cm}

8\times225\text{cm}= 1,800\text{cm}

We then need to add up all these new values. This will give us the estimated total jump length for all of the jumps completed.

50\text{cm}+1,350\text{cm} +7,000\text{cm} +5,600\text{cm} +1,800\text{cm} =15,800\text{cm}

This means that the total estimated jump length was 15,800cm. By adding up the frequency column, we know that 118 jumps were completed (4+18+56+32+8=118\text{ jumps}). Since there were 118 jumps with an estimated total jump length of 15,800cm, then the estimated mean can be calculated as follows:

\dfrac{15,800 \text{cm}}{118\text{ jumps}} = 134 \text{cm to the nearest cm}

3) Below is a grouped frequency table of data collected on the lengths of students’ journeys to school:

To the nearest minute, find an estimate for the mean of this data.

The first thing we need to do is add an additional midpoint column to our table. Since we have a range of journey times, we are assuming that all journeys are exactly in the middle of the range. This is probably not the case in reality, which is why this is an estimated mean rather than the actual mean.

The first midpoint, the point which is half-way between 0 minutes and 10 minutes, is 5 minutes. If you are unsure how to calculate a midpoint, add up the two values and divide by 2. For example, working out the midpoint in the 32 – 45 minute range is not that easy::

\dfrac{32 + 45}{2}=38.5

Once you have calculated all the midpoints, you should have a table that looks like the table below:

We then need to multiply the frequency of each group by its midpoint. This will give us the estimated total journey time for the number of journeys completed within each range of journey time.

12\times5\text{ minutes}= 60\text{ minutes}

18\times12\text{ minutes}= 216\text{ minutes}

34\times17\text{ minutes}= 578\text{ minutes}

33\times260\text{ minutes}= 858\text{ minutes}

19\times38.5\text{ minutes}= 731.5\text{ minutes}

We then need to add up all these new values. This will give us the estimated total journey time for all of the journeys completed.

60\text{ minutes}+216\text{ minutes}+578\text{ minutes}+858\text{ minutes}+731.5\text{ minutes} = 2443.5\text{ minutes}

This means that the total estimated journey time was 2443.5 minutes. By adding up the frequency column, we know that 116 journeys were completed (12+18+34+33+19=116\text{ journeys}). Since there were 116 journeys with an estimated total time of 2443.5 minutes, then the estimated mean can be calculated as follows:

\dfrac{2443.5 \text{ minutes}}{116\text{ journeys}} = 21 \text{ minutes to the nearest minute}

4) Simon and Steve went on a 2-day fishing expedition and took note of the lengths of all the fish they caught. The data is shown in the table below:

a) To the nearest cm, find an estimated mean for the data.

b) What would the estimated mean be if the 6 biggest fish were not included in the data?

a) The first thing we need to do is add an additional midpoint column to our table. Since we have a range of fish size lengths, we are assuming that all fish caught fall exactly in the middle of the range. This is probably not the case in reality, which is why this is an estimated mean rather than the actual mean.

The first midpoint, the point which is half-way between 0 cm and 20 cm, is 10 cm. If you are unsure how to calculate a midpoint, add up the two values and divide by 2. For example, working out the midpoint in the 90 – 300 cm is not that easy:

\dfrac{90 + 300}{2}=195

Once you have calculated all the midpoints, you should have a table that looks like the table below:

We then need to multiply the frequency of each group by its midpoint. This will give us the estimated total fish length for the number of fish caught within each range of fish length.

16\times10\text{cm}= 160\text{cm}

27\times25\text{cm}= 675\text{cm}

9\times40\text{cm}= 360\text{cm}

13\times60\text{cm}= 7,800\text{cm}

8\times80\text{cm}= 640\text{cm}

6\times195\text{cm}= 1,170\text{cm}

We then need to add up all these new values. This will give us the estimated total length for all of the fish caught.

160\text{cm}+675\text{cm}+360\text{cm}+7,800\text{cm}+640\text{cm}+1,170\text{cm}=10,805\text{cm}

This means that the estimated total length of all the fish caught was 10,805cm. By adding up the frequency column, we know that 79 fish were caught (16+27+9+13+8+6=79\text{ fish caught}.) Since there were 79 fish caught with an estimated combined length of 10,805cm, then the estimated mean can be calculated as follows:

\dfrac{10,805 \text{cm}}{79\text{ fish}} = 137\text{cm to the nearest cm}

b) If we do not take into consideration the 6 fish that were in the 90 – 300 cm category, then the total number of fish caught is reduced from 79 to 73.

The estimated total length of all the 79 fish caught was 10,805cm. We need to subtract the 6 fish in the 90 – 300cm category:

10,805\text{cm} - (6 \times 196\text{cm}) = 9,629 \text{cm}

Therefore, excluding the 6 biggest fish, the estimated total length of the other 73 fish was 9,629cm. The estimated mean can therefore be calculated as follows:

\dfrac{9,629 \text{cm}}{73\text{ fish}} = 132\text{cm to the nearest cm}

5) Suzanna and Floella are organising a charity event which involves throwing darts at a board to see how many times a participant can hit the bullseye in a 10-minute period.

Suzanna compiles the data in the table below:

a) According to Suzanna, what is the estimated mean number of bullseyes hit in the 10-minute period. Give your answer to the nearest whole number.

b) Floella compiles the data differently to Suzanna by organising the data in batches of 20 hits instead of batches of 10 hits. What does Floella’s data table look like and how does it affect the estimated mean when she calculates it to the nearest whole number?

a) The first thing we need to do is add an additional midpoint column to our table. Since we have a range of number of bullseyes hit, we are assuming that the number of bullseyes hit by each participant falls exactly in the middle of the range. This is probably not the case in reality, which is why this is an estimated mean rather than the actual mean.

The first midpoint, the point which is half-way between 0 cm and 10 cm, is 5 cm. If you are unsure how to calculate a midpoint, add up the two values and divide by 2. In this table, the midpoints are all quite easy to calculate .

Once you have calculated all the midpoints, you should have a table that looks like the table below:

We then need to multiply the frequency of each group by its midpoint. This will give us the estimated total number of bullseyes hit by participants who fall into this category.

3\times5\text{ hits}= 15\text{ hits}

25\times15\text{ hits}= 375\text{ hits}

28\times25\text{ hits}= 700\text{ hits}

19\times35\text{ hits}= 665\text{ hits}

8\times45\text{ hits}= 360\text{ hits}

2\times55\text{ hits}= 110\text{ hits}

We then need to add up all these new values. This will give us the estimated total number of times the bullseye was hit.

15\text{ hits}+375\text{ hits}+700\text{ hits}+665\text{ hits}+360\text{ hits}+110\text{ hits}=2,225\text{ hits}

This means that the estimated total number of bullseyes hit was 2,225. By adding up the frequency column, we know that there was a total of 85 participants (3+25+28+19+8+2=85\text{ participants}.) Since there were 85 participants with an estimated total of 2,225 bullseyes hit, then the estimated mean can be calculated as follows:

\dfrac{2,225 \text{ bullseyes hit}}{79\text{ participants}} = 26\text{ hits to the nearest whole number}

b) By organising the data in batches of 20 hits, rather than batches of 10 hits, each row in Floella’s table will combine 2 of Suzanna’s rows, so the table will be half the size. It will look as follows:

In order to calculate the estimated mean, we will need to work out new midpoints, as follows:

Using Floella’s table, the estimated mean can be calculated as follows:

\dfrac{(10\times28) + (47\times30) + (10\times 50)}{85}=26\text{ hits}

When the answer is rounded to the nearest whole number, the estimated mean is the same.

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