Averages and Range

Range

Range = highest amount - lowest amount

Types of average

– Sometimes we are told that the average family has 2.4 children.

– There are three types of average: mode, median and mean

• Mean• If amounts are shared out

equally, then each person (or item) will get the same amount which is the mean. If the total number of children are shared out between all the families then there are 2.4 children per family.

ModeThe mode is the most common amount. Most families have 2 children so the mode of the number of children is 2.

MedianWhen the amounts are arranged in order, the middle one is the median.

– A die is tossed 15 times and the scores were:– 6, 5, 4, 3, 4, 1, 1, 2, 4, 4, 3, 2, 3, 5, 4

– Score Frequency– 1 2– 2 2– 3 3– 4 5– 5 2– 6 1

From the table we can see that the score with the highest frequency is 4, so the mode is 4.

– Writing the scores in order gives– 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 6

– The median is the middle one i.e. 4

– The mean is found by adding together all the scores and dividing by the number of scores (15)

– 1+1+2+2+3+3+3+4+4+4+4+4+5+5+6 = 51

3.4 1551

Mean

The mean height of this mountain range is 1000m

1000 m

The mean height of this mountain range is also 1000m. The mean can mask extreme values.

1000m

Mean, Mode and Median of Frequency Distributions

– The ages of 50 people in a maths club are given below

– Find the mean, mode and median of their ages.•

– 23 has the highest frequency, so the mode is 23•

– where f is the total of the frequencies and fx is the total of the frequencies ×

values

Age 20 21 22 23 24 25 26 27 28 29 30

Frequency 3 5 4 8 3 6 7 4 4 4 2

f

fx x

– To find the mean, rewrite the frequency table as shown

Age(x)

Frequency(f)

Frequency × age(f × x)

2021222324252627282930

35483674442

y 24.74 50

1237

f

fx

mean60

105

88

184

72

150

182

108112

11660

50 1237

– To find the mean and median, put the data into a frequency table.

Number of children (x)

Number of families (f) f × x

0123

515105

0152015

35 50

1.43 3550

ffx

Mean

18th 2

36

21 35

position Median

The first 5 families have 0 children, the next 15 families have 1 child. Therefore the 18th family has 1 child.

Grouped Data– For a grouped frequency distribution the modal

class is the group (or class) with the highest frequency.

– An estimate for the mean can be found by assuming that each value in a class is equal to the value of the mid-point of the class and then using the formula

f

fx

frequency

midpoint frequency Mean

• Example– At a supermarket, the manager recorded the lengths of time

that 80 customers had to wait in the check-out queue.– Find the modal class and calculate an estimate for the

mean.

Waiting time (t) Mid-pointx

Frequencyf f × x

0 ≤ t < 5050 ≤ t < 100

100 ≤ t < 150150 ≤ t < 200200 ≤ t < 250

250 ≤ t < 300

2575

125175225275

58

10162813

125600

1250280063003575

80 14650

– The class with the highest frequency is the 200 ≤ t < 250 class. The modal class is 200 ≤ t < 250

– The mean is

Waiting time (t) Mid-pointx

Frequencyf f × x

0 ≤ t < 5050 ≤ t < 100

100 ≤ t < 150150 ≤ t < 200200 ≤ t < 250

250 ≤ t < 300

2575

125175225275

58

10162813

125600

1250280063003575

80 14650

183.1 80

14650

ffx

Histograms• Histograms are used to represent information

contained in grouped frequency distributions.

• The horizontal axis is a continuous scale• There are no gaps between the bars

Histogram yn dangos Taldra Bechgyn Histogram representing Boys’ Heights

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

1

2

3

4

5

x

f

Histogram yn dangos Taldra GenethodHistogram representing Girls’ Heights

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

1

2

3

4

5

x

f

Frequency Polygons• Frequency polygons are often used instead of

histograms when we need to compare two or more groups of data

• To draw a frequency polygon:Plot the frequencies at the midpoint of each class interval Join successive points with straight lines

• To compare data, frequency polygons for different groups of data can be drawn on the same diagram

Polygon Amlder a Histogram o Daldra BechgynFrequency Polygon and Histogram of Boys Heights

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

1

2

3

4

5

x

f

Polygon Amledd a Histogram o Daldra GenethodFrequency Polygon and Histogram of Girls Heights

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

1

2

3

4

5

x

f

Defnyddio polygonau amledd i gymharu dwy set o ddata

Using frequency polygons to compare two sets of data

1.3 1.4 1.5 1.6 1.7 1.8 1.9 20

1

2

3

4

5

6

x

f

Boys Heights (m)

Girls Heights (m)