A coffee shop conducted a two-day survey to determine the average number of cappuccinos made per hour. A histogram showed the frequency of cappuccinos made within various hourly intervals. To calculate the average, interval midpoints were determined and multiplied by the frequencies. The total of these products was divided by the total frequency, determining that the average number of cappuccinos made per hour was 10.

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0 - 3 2
4 - 7 3
8 – 11 8
12 - 15 3
16 - 19 2
Number of
Cappuccinos
Freq
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2. 0 - 3 2
4 - 7 3
8 – 11 8
12 - 15 3
16 - 19 2
Number of
Cappuccinos
Freq
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A survey was conducted at a Cafe which sells food and coffees.
The Barista suggested that they get a bigger machine to cope with
high demand. A bigger machine is very expensive to buy, and so the
owner had a two day survey done to find out how many Cappuccinos
were being made per hour in the Cafe.
From the survey results, they would be able to do some Statistics,
and better understand the current problem situation.

3. The following is a Histogram of the Number of Coffees made
for each hour that the coffee shop was open over two days.
We want to find the Average Number of Coffees per hour.
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0 - 3 2
4 - 7 3
8 – 11 8
12 - 15 3
16 - 19 2
Number of
Cappuccinos
Freq
0–3
8
6
4
2
0
Number of Cappuccinos Made Per Hour

4. The “Modal Class” is the group which occurs the most times.
It is the biggest frequency group and forms tallest column.
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0 - 3 2
4 - 7 3
8 – 11 8
12 - 15 3
16 - 19 2
Number of
Cappuccinos
Freq
0–3
8
6
4
2
0
Number of Cappuccinos Made Per Hour
The “Modal Class” is the group 8 to 11 Cappuccinos.

5. There can be several groups in the “Modal Class” if we have
the biggest frequency value occur multiple times.
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0 - 3 2
4 - 7 3
8 – 11 3
12 - 15 3
16 - 19 2
Number of
Café Lattes
Freq
0–3
8
6
4
2
0
Number of Café Lattes Made Per Hour
“Modal Class” is 4-7, 8-11, and 12-15 Café Lattes.

6. The Median is halfway through our Groups of Interval Data.
Cappuccinos f
0-3 2
4-7 3
8-11 8
12-15 3
16-19 2
TOTAL 18
We find the Median by
finding our Half-Way
Position within the
Frequencies, which is
NOT 18 / 2 = 9 
1,2,3,4,5,6,7,8,9, 10,11,12,13,14,15,16,17,18
The middle of our Frequencies is at the 9.5th value.
9.5

7. The Median is halfway through our Groups of Interval Data.
Cappuccinos f
0-3 2
4-7 3
8-11 8
12-15 3
16-19 2
TOTAL 18
We can find our Half-Way
Position using the formula:
(Total Frequency + 1) / 2
= (18 + 1) / 2 = 9.5
1,2,3,4,5,6,7,8,9, 10,11,12,13,14,15,16,17,18
The middle of our Frequencies is at the 9.5th value.
9.5

8. To find the Median, we also need to do Cumulative Addition.
Cappuccinos f Cumulative f
0-3 2
4-7 3
8-11 8
12-15 3
16-19 2
TOTAL 18
We add a new column
where we do continual
Totalling of the “f”
Frequency values.
We add on an extra column, and add the frequencies.
But first we need to learn how to do Cumulative Addition.

9. Our friend Glen is busking and getting money for each song.
Song $ Cumulative $
Let It Be 15 15
Get Back 4 15 + 4 19
Lady
Madonna
10 15 + 4 + 10 29
Imagine 20 15+4+10+20 49
TOTAL 49
After he has finished each song, we can see how
many total dollars he has. The last values must be
identical (eg. both 49), or else we have made an error.

10. A quicker way to add up is using the “Zig Zag” pattern.
Song $ Cumulative $
Let It Be 15 15
Get Back 4 19
Lady
Madonna
10 29
Imagine 20 49
TOTAL 49
After he has finished each song, we can see how
many $ he has earned so far. The last values must be
identical (eg. both 49), or else we have made an error.
+
+
+

11. 1st step is add the extra column, and do Cumulative Addition.
Cappuccinos f Cumulative f
0-3 2 2
4-7 3 5
8-11 8 13
12-15 3 16
16-19 2 18
TOTAL 18
We add a new column
where we do continual
Totalling of the “f”
Frequency values.
(Use Zig Zag Pattern)
The other step is to find the half-way frequency using
( Total f + 1 ) / 2 = (18 + 1 ) / 2 = 9.5th position.

12. Half-Way through the Frequency is (18 + 1) /2 = 9.5
Cappuccinos f Cumulative f
0-3 2 2
4-7 3 5
8-11 8 13
12-15 3 16
16-19 2 18
TOTAL 18
The 9.5th position
occurs during
the 8-11 Interval.
In this Interval
we pass through
9.5 on the way
to reaching 13.
The Median Class is 8-11

13. We find the Average Number of Coffees per Hour by adding
two new columns to our Frequency Table and using a Formula.
Cappuccinos Freq Interval Midpoint Freq x Midpt
0-3 2
4-7 3
8-11 8
12-15 3
16-19 2
TOTALS 18
We find all of our Interval Midpoints first, and then we
Multiply each Frequency x Midpoint value as our second step

14. We find the Average Number of Coffees per Hour by adding
two new columns to our Frequency Table and using a Formula.
Cappuccinos Freq Interval Midpoint
0-3 2 0,1, 2,3 -> 1.5
4-7 3 4,5, 6,7 -> 5.5
8-11 8 8,9, 10,11 -> 9.5
12-15 3 12,13, 14,15 13.5
16-19 2 16,17, 18,19 17.5
TOTALS 18
We find Interval Midpoints by getting the exact middle
(or Median) of each of our data classes, or data groups.
Eg. Exactly half way between 0 to 3 is the value of 1.5

15. We can also find the Midpoints by adding the Lower end
to the Upper end of an Interval, and then dividing by 2.
Cappuccinos Freq Interval Midpoint
0-3 2 (0 + 3) /2 -> 1.5
4-7 3 (4 + 7) /2 -> 5.5
8-11 8 (8+11) /2 -> 9.5
12-15 3 (12+15) /2 -> 13.5
16-19 2 (16+19) /2 -> 17.5
TOTALS 18
We find Interval Midpoints by getting the exact middle
(or Median) of each of our data classes, or data groups.
Eg. Exactly half way between 0 to 3 is the value of 1.5

16. We complete the final column by Multiplying the Frequency
by the Midpoint Value for each row of our class groups.
Cappuccinos Freq Interval Midpoint Freq x Midpt
0-3 2 0,1,2,3 -> 1.5 2 x 1.5 = 3
4-7 3 4,5,6,7 -> 5.5 3 x 5.5 = 16.5
8-11 8 8,9,10,11 -> 9.5 8 x 9.5 = 76
12-15 3 12,13,14,15 13.5 3 x 13.5 = 40.5
16-19 2 16,17,18,19 17.5 2 x 17.5 = 35
TOTALS 18 171
The 3rd and final step is to use the Grouped Mean Formula:
MEAN Average = Total of (Freq x Midpt) / Total Frequency

17. We find the Average Number of Coffees per Hour by adding
two new columns to our Frequency Table and using a Formula.
Cappuccinos Freq Interval Midpoint Freq x Midpt
0-3 2 1.5 2 x 1.5 = 3
4-7 3 5.5 3 x 5.5 = 16.5
8-11 8 9.5 8 x 9.5 = 76
12-15 3 13.5 3 x 13.5 = 40.5
16-19 2 17.5 2 x 17.5 = 35
TOTALS 18 171
MEAN Average = Total of (Freq x Midpt) / Total Frequency
= 171 / 18 = 10 cappuccinos per hour

18. We use some specific Mathematical Notation as follows:
Groups like 0-3 are called “Intervals” eg. 0 up to 3 = 0,1,2,3
f = frequency, or how many we have of a particular interval
x = midpoint, which is (Low + High) /2 for each interval
is the sum of the frequencies (eg. Total Frequency)
is the Total of all the Frequency x Midpoint values
MEAN Average for Grouped Data =
f
fx
f
fx

19. Using Standard Statistics Notation, we obtain the following:
Cappuccinos f x f x
0-3 2 1.5 2 x 1.5 = 3
4-7 3 5.5 3 x 5.5 = 16.5
8-11 8 9.5 8 x 9.5 = 76
12-15 3 13.5 3 x 13.5 = 40.5
16-19 2 17.5 2 x 17.5 = 35
TOTALS = 18 = 171
MEAN = = 171 / 18 = 10 cappuccinos per hour
f fx
f
fx

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