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Mathematics

Data Handling - Basic Statistics

Averages

The number chosen for representing a set of data is called its average. It must,

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

There are three averages:

  1. Mean
  2. Mode
  3. Median

Mean

This is the sum of all values divided by the number of data - Σ,sigma, means, add
Mean = ∑ x / n

Mode

This is the value that occurs most frequently.

Range of Data

The difference between the highest and lowest values is called the range of the data.

E.g. Data: 3, 4, 8, 6, 11, 21
Range = 21 - 3 = 18

Median

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

Finding Averages of Raw Data

Data that exists in the exact form since its collection is usually considered raw data.

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.


Finding Averages of Tabular Data

Raw data, when arranged in a table for convenience, is in tabular form.

E.g.

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 = sum of fx/n ∑ fx / n = 3X3 + 4X5 + 5X10 + 6X8 + 7X4 /30 = 5.2
TheMedian Class is the class where (n/2)th 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.

Finding Averages of Grouped Data

Raw data, when arranged in classes for easy handling, form grouped data. It is usually called a group frequency table.

E.g.

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 values 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!

Interactive Practice

With the following applet, you can practise the averages of a randomly-generated set of raw data:

Practice Questions

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)
    305
    40k
    501

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

  5. The median of five consecutive odd numbers is M. Find the mean of the numbers in terms of M. 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.
  8. Hounslow United football club in West London managed to score 2, 4, 4, 4, 2, 1, 4 goals in their first 7 matches. Find the mean.The manager wants them to keep the average goal of the first 10 matches to be the same as the mean goal so far. How many goals should they score in the next three matches?
  9. The mean of 6 numbers is 8. The mean of two numbers is 5. Find the mean of the other four numbers.
  10. The list of numbers, 2, 3, 7, 8, x, 12, 16, is in ascending order. Its mode, median and mean are the same. Find x.

 

Answers

Move the mouse over, just below this, to see the answers:

  1. 3.4. 3, 3
  2. 5.5, 5, 5
  3. 1.53
  4. 7
  5. (M + 4), (M + 4)2 + 8
  6. 12, 18
  7. 54
  8. 3, 9
  9. 10
  10. 8

 

Now that you have read this tutorial, you will find the following tutorials very helpful too: