FORMULATION OF FREQUENCY TABLE FOR UNGROUPED DATA
UNGROUPED DATA: Ungrouped data is one in which the raw data has occurrences or frequencies more than and are without class intervals. In the formulation of frequency table for ungrouped data, two basic steps are taken.
- Prepare a tally sheet.
- Prepare a frequency table.
- PREPARATION OF A TALLY SHEET: This is when the variables are taken one after the other with a stroke called tally. The tally of five makes a bundle i.e.
- PREPARATION OF A FREQUENCY TABLE: The frequency table is simply obtained by adding the tallies together in a separate column referred to as frequency.
Example: The following are scores of thirty (30) students of SS 1 in an economics test.
2, 4, 8, 8, 2, 6, 6, 8, 2, 4
8, 0, 8, 6, 0, 10, 2, 2, 0, 10
4, 6, 0, 10, 2, 2, 6, 6, 4, 2
| Scores | Tally | Frequency |
| 0 | Illl | 4 |
| 2 | llll lll | 8 |
| 4 | Illl | 4 |
| 6 | llll l | 6 |
| 8 | llll | 5 |
| 10 | Ill | 3 |
| 30 |
MEASURES OF CENTRAL TENDENCY
- Mean
- Median
- Mode
MEASURES OF CENTRAL TENDENCY
Measures of central tendency means are values which show the degree to which a given data or any given set of values will converge toward the central point of the data. It is also called measure of location and is the statistical information that gives the middle or centre or average of a set of data. It includes mean, median and mode.
THE MEAN
Mean or arithmetic mean is defined as the sum of series of figures divided by the number of observations. It is the commonest and the most widely used among the other types of averages or measures of central tendency.
TYPES OF MEAN
- The Arithmetic Mean
- The Geometric Mean
- The Quadratic Mean
Example
Calculate the arithmetic mean of the following scores of eight students in an economics test. The scores are: 14, 18, 24, 16, 30, 12, 20, and 10.
Solution
Add up the scores
14+18+24+16+30+12+20+10 = 144
Number of observation (students) = 8
Arithmetic Mean =Sum of observations divided by Number of observations
= 144 = 18
8
ADVANTAGES OF THE MEAN
- It is easy to derive or calculate.
- It is easy to interpret.
- It is the best known average.
- It has determinate exact value.
- It provides a good measure of comparison.
DISADVANTAGES OF THE MEAN
- It is difficult to determine without calculation.
- Some facts may be concealed.
- It cannot be obtained graphically.
- If one or more value is incorrect or missing, calculation becomes difficult.
- It may lead to distorted results.
EVALUATION
- Define mean.
- What are the disadvantages of mean?
THE MEDIAN
The median is an average which is the middle value when figures are arranged in their order of magnitude either in ascending or descending order, especially from ungrouped data.
Example 1:
Calculate the median of the following scores: 12, 8 15, 9, 3, 7, and 1
Solution
Step 1: First arrange in order
1, 3, 7, 8, 9, 12 and 15
Step 2: Total frequency is 7, thus the middle number in the set is in the 4th position
Step 3: Median = 4th Position = 8
Example 2:
Find the median of this set of numbers; 36, 42, 10, 15, 9, 32 16 and 12.
Solution
Step 1: 9, 10, 12, 15, 16, 32 36 and 42
Step 2: Total Frequency = 8
Step 3: Median = 4th and 5th Position
= 4th + 5th = 15 + 16
2
= 31 = 15.5
2
Example 3
The following are scores of 20 students in an Economics test. What is the median Mark?
5 10 2 9 5 3 4 6 1 3
2 3 6 1 3 3 2 3 4 3
Solution
| Marks | Tally | Frequency |
| 1 | II | 2 |
| 2 | III | 3 |
| 3 | III I | 6 |
| 4 | II | 2 |
| 5 | III | 3 |
| 6 | II | 2 |
| 9 | I | 1 |
| 10 | I | 1 |
| 20 |
Median = f = 70th + 11th = 3 + 3 = 6 = 3
x 2 2 2
ADVANTAGES OF THE MEDIAN
- It is easy to determine with little or no calculations
- It is easy to understand and compute
- It does not use all values in the distribution
- It gives a clean idea of the distribution
DISADVANTAGES OF THE MEDIAN
- It is not useful in further statistical calculation
- It ignores very large or small values
- It does not represent a true average of the set of data.
THE MODE
This is the most frequently recurring number in a set of numbers or data, that is to say, it is the number or value with the highest frequency. It tells us the observation which is most popular. The best and easiest way of calculating the mode of any distribution is to form a frequency table for it.
Example
Using the frequency distribution of example 3 above, the mode is 3 because it has the highest frequency of six (6)
MERITS
- It can be easily understood.
- It is not affected by extreme values.
- Easy to calculate from the graph.
- It is easy to determine.
DEMERIT
- It can be a poor average.
- It can be difficult to compute if more than one mode exists.
- It is not useful in further statistical calculations.
- All the values used in the distribution are not considered.
EVALUATION
- What is median?
- Write a short note on measures of central tendency.
REVISION/GENERAL EVALUATION
- Find the mean of the following set of numbers 14,11,12,13,11,14,12,20,24,21,22,23,20,11,13,23.
- Define the median. State four advantages of median.
- Calculate the mean 18,14,14,15,13,18,19,19,19,21.
- What are the disadvantages of the mean.
- Define the mode.
- List three advantages of the mode
WEEKEND ASSIGNMENT
- A stroke of five (5) makes up a (a) frequency (b) tally (c) bond (d) bundle
- The measure of central tendency in which the sum of observations is divided by the number of observations is called……. (a) median (b) mean (c) mode (d) range
- A data with figures randomly given without any sort of arrangement is called…………. (a) array of data (b) rough data (c) treated data (d) raw data
Use the table below to solve the following questions
| Meat Purchased (kg) | 1 | 2 | 3 | 4 | 5 | 6 |
| Frequency | 2 | 4 | 10 | 8 | 5 | 1 |
- The mean score is ________ (a) 4.34 (b) 3.34 (c) 3.43 (d) 4. 33
- The modal score is (a) 2 (b) 3 (c) 4 (d) 5
- The median score is ___________ (a) 2 (b) 3 (c) 4 (d) 5
SECTION B
- What are the advantages of median?
- List three disadvantages of mean.
See also