Data Handling by Example

Averages

  • Represent all the values
  • Should not be an exaggerated one - not too small or not too big

We call this special value, the average.


There are three averages:

  1. Mean
  2. Mode
  3. Median

Mean

This is the sum of all values divided by the number of data.
Mean = ∑ x / n

Mode

This is the value that occurs most frequently.

Median

This is the middle value, when the data is arranged in order of size. Now, let's try some examples.

E.g.1

The heights of five plants in a garden are 3cm, 4cm, 7cm, 12cm and 9cm. Find the averages.

Mean = ∑ x / n = 3 + 4 + 7 + 12 + 9 / 5 = 7cm
There is no mode, as each value occurs only once.
To find the median, let's rearrange them in order of size:
3, 4, 7, 9, 12
The middle value is 7. So, the median = 7cm.

E.g.2

The lengths of 6 carpets are 7m, 15m, 15m, 9m, 22m, 4m. Find the averages.

Mean = ∑ x / n = 7 + 15 + 15 + 9 + 22 + 4 / 6 = 12m
The mode = 15m
To find the median, let's rearrange them in order of size:
4, 7, 9, 15, 15, 22
The middle value = 9 + 15 /2 = 12, and so is the median.


E.g.3

The frequency of shoe sizes of students in a certain class is as follows:

shoe-size (x)frequency (f)
33
45
510
68
74

Here, we have a slightly different approach;
Mean = ∑ fx / n = 3X3 + 4X5 + 5X10 + 6X8 + 7X4 /30 = 5.2
TheMedian Class is the class where n/2 the value lies in. In this case, 30/2 = 15th value lies in shoe-size 5 class. So, it is the median class.
The Modal class is the class with the highest frequency. So, the modal class is shoe-size 5 class.

E.g.4

The marks obtained by a group of students for maths are as follows:

Marks (x)frequency (f)
0 - 203
21 - 406
41 - 609
61 - 808
81 - 1004

Mean = ∑ fx / n = 10X3 + 30X6 + 50X9 +70X8 + 90X4 /30 = 52.7 - x is the middle class value
The Median Class is the class where n/2 the value lies in. In this case, 30/2 = 15th value lies in 41 - 60 class. So, it is the median class.
The Modal class is the class with the highest frequency. So, the modal class is 41 - 60 class.

The reliability of the Mean

The mean can easily be influenced by the extremes of data:

E.g

The heights of five plants are 2cm, 4cm, 7cm, 18cm, 19cm. Find the mean and comment on the result.
Mean = 2 + 4 + 7 + 18 + 19 / 5 = 10 cm
This value does not represent either the shortest plant - 2cm - or the tallest - 19cm. So, the mean in this case is not accurate; it may even mislead!

If you would like to practise more, please visit this page.

Now, in order to complement what you have just learnt, work out the following questions:

  1. Find the mean, median and mode of the following numbers - 1, 5, 3, 4, 3, 8, 2, 3, 4, 1.
  2. The marks scored by pupils in a certain class for an IQ test are as follows:

    Marks (x)frequency (f)
    32
    43
    56
    64
    73
    82

    Find the averages of the marks.

  3. The mean height of 4 boys is 1.2m and the mean height of 6 girls is 1.5m. Find the mean height of 10 pupils altogether.
  4. The marks for a certain test for a group of students are as follows:

    Marks (x)frequency (f)
    302
    40k
    501

    The mean mark for the group is 30. Find k.

  5. The median of five consecutive odd numbers is T. Find the mean of the numbers in terms of T. Hence, find the mean of the square of the same numbers.
  6. The numbers 4, 5, 9, 15 and k are arranged in ascending order so that the mean is the same as median. Find k. Without further calculation, determine the new mean if the numbers are doubled.
  7. A set of numbers are in the ratio 3: 5: 8: 12. The mean turns out to be 42. Find the range of the numbers.

 

 

 

 

Resources at Fingertips

This is a vast collection of tutorials, covering the syllabuses of GCSE, iGCSE, A-level and even at undergraduate level. They are organized according to these specific levels.
The major categories are for core mathematics, statistics, mechanics and trigonometry. Under each category, the tutorials are grouped according to the academic level.
This is also an opportunity to pay tribute to the intellectual giants like Newton, Pythagoras and Leibniz, who came up with lots of concepts in maths that we take for granted today - by using them to serve mankind.

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Recommended Reading

 

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