Measures Of Central Tendency
Statistics is the collection and summary of data from the world around us. Data are collected and analysed in order to make predictions or to draw conclusions about a certain area of interest.
We often describe a set of data by choosing a single number that indicates where the data in the set are centred or concentrated. This is called the central tendency of the data. This number may indicate the values that occurs the most often, or the values that lie closest to the middle of the set of data. You see, in this post, we will revise the concepts of mean, median and mode, how to calculate the mean from a frequency distribution table and how to use the range as a measure of spread.
In statistics, data are divided into three types: raw data, qualitative data and quantitative data.
Types of data
1. Raw data
Raw data are not organised in any particular order, but are made up of information that researchers obtain in the field (for example, by directly observing people and events) before they sort or analyse what they are seeing.
2. Qualitative data
Qualitative data can be described in words only, and cannot be counted. Examples of qualitative data are colours and feelings.
3. Quantitative data
Quantitative data consist of numerical values obtained by counting objects, events and so on. Quantitative data can either be discrete or continuous.
Now that we know the types of data. But do we know how we sort quantitative data in a way that is easier to read and retrieve? This process is called Frequency distribution table.
Frequency Distribution Table
A frequency distribution table has three columns. The first column contains the primary data events/numbers, usually in ascending order of magnitude. The second column contains the tallies of the individual data items. The third contains the frequency (how often does the number occur)
Example
The shoe sizes of 30 students are listed below.
Construct a frequency table to show the distribution of shoe sizes.
Solution
Types of measures of central tendency
We use three measures of central tendency:
The mean, the median and the mode.
1. The mean
The mean is the average of the data set and takes into account all the values in the data set. We calculate the mean by adding all the values in the data set and dividing it by the total number of values. Note that the mean is affected by outliers. Outliers are data values that are very low or very high. They affect the mean by inflating or deflating it.
2. The median
When data is arranged in ascending order, it is arranged from the smallest value to the biggest value (for example, 2;3;4;5;6). When data is arranged in descending order, it is arranged from the biggest value to the smallest value (for example, 6;5;4;3;2). The middle value in a data set that is arranged in ascending order is called the median.
3. The mode
The mode is the data value that appears most often in a set of data. We do not need to calculate the mode. We only need to find the value that appears most frequently. For example, if you have numbers 2;5;7;7;10;12;15, the mode is 7. If no number is repeated, then there is no mode for the list. You must also be aware that there can be more than one mode.
Example
Find the mean, mode and median of each of these sets of data.
(1). 11, 16, 17, 13, 16, 13, 18, 19, 14, 16, 12
(2). 10, 34, 20, 19, 35, 20, 25, 23, 15, 25, 26, 11, 27, 32, 17
(3). 4, 11, 5, 9, 17, 16, 8, 10
Solutions
(1) 11, 16, 17, 13, 16, 13, 18, 19, 14, 16, 12
Mean: To find mean, count the number of primary data given. In this case we have 11 (i.e n = 11). Add all the numbers together and divide it by n.
n = 11
= 11+16+17+13+16+13+18+19+14+16+13
= 165
= sum/n
= 165/11
= 15
Mode: The mode is the value that occurs most frequently in the set. In this set, 16 occurs more times than any other number.
Therefore mode = 16.
Median: The median is the middle value in the set of data arranged in ascending order, i.e
11,12,13,13,14,16,16,16,17,18,19.
The middle value here is 16.
NB: Not just because it occurs the most. But because of it's the middle number.
(2). 10, 34, 20, 19, 35, 20, 25, 23, 15, 25, 26, 11, 27, 32, 17
Mean:
n = 15
Sum = 10,34,20...+32
= 339
Mean = sum/n
= 339/15
= 22.6
Mode: 20 and 25 both occur most frequently (twice). There are two modes i.e the data set is bimodal.
Therefore, mode = 20, 25
Median:
10,11,15,17,19,20,20,23,25,25,26,27,32,34,35
The middle value is 23
(3) 4, 11, 5, 9, 17, 16, 8, 10
Mean:
n = 8
Sum = 4+11+5+9+17+16+8+10
= 80
Mean = sum/n
= 80/8
= 10
Mode: No number occur more than each other, i.e no repeated numbers. Hence, no mode.
Median: 4,5,8,9,10,11,16,17
There are two middle values here. The median is the average of these two numbers. So what you do here is add the 2 numbers and divide by 2. That is, (9 + 10)/2
= 19/2
= 9.5
Can you find the mode of this set of numbers?
18, 24, 24, 16, 31, 18, 15
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