MEASURES OF CENTRAL TENDENCY AND LOCATION: MEAN, MODE, MEDIAN AND GRAPHICAL LOCATION OF MODE, MEDIAN, QUARTILES, DECILES AND PERCENTILES

Measures of Central Tendency:

1. Mean: The mean is the average of a set of values. It’s calculated by adding up all the values and dividing by the total number of values.
2. Mode: The mode is the value that appears most frequently in a dataset. A dataset can have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode at all.
3. Median: The median is the middle value of a dataset when the values are arranged in order. If there’s an odd number of values, the median is the middle one. If there’s an even number of values, the median is the average of the two middle values.

Graphical Location:

1. Graphical Location of Mode: The mode can be found as the peak or highest point on a histogram or frequency distribution graph.
2. Graphical Location of Median: The median divides the data into two equal halves. On a histogram, it’s where the distribution curve would be divided into two equal areas.

Quantiles and Deciles:

1. Quartiles: Quartiles divide the data into four equal parts. Q1 is the value below which 25% of the data falls, Q2 is the median (50th percentile), and Q3 is the value below which 75% of the data falls.
2. Deciles: Deciles divide the data into ten equal parts. D1 is the value below which 10% of the data falls, D5 is the median, and D9 is the value below which 90% of the data falls.

Percentiles:

Percentiles divide the data into 100 equal parts. P1 is the value below which 1% of the data falls, P50 is the median, and P75 is the value below which 75% of the data falls.

Graphical Representation of Quantiles and Percentiles:

On a cumulative frequency curve (ogive), quartiles, deciles, and percentiles are marked as points where the cumulative frequency crosses certain percentage levels.

These measures help to understand the central location and spread of data. The mean, mode, and median provide insight into the typical value, while quartiles, deciles, and percentiles offer information about the data’s distribution and variation. Graphical representation makes it easier to visualize these concepts in a dataset

[mediator_tech]

1. The _____ is calculated by adding up all values and dividing by the total number of values.
(a) mode
(b) median
(c) mean

2. The mode is the value that appears _____ frequently in a dataset.
(a) least
(b) most
(c) equally

3. The median is the middle value of a dataset, and it separates the data into _____ halves.
(a) unequal
(b) two equal
(c) three equal

4. The mode can be graphically located at the _____ of a histogram.
(a) peak
(b) base
(c) center

5. The graphical location of the median is where the distribution curve is divided into _____ areas.
(a) uneven
(b) three
(c) two equal

6. Quartiles divide the data into _____ equal parts.
(a) three
(b) four
(c) five

7. The value below which 25% of the data falls is called the _____ quartile.
(a) 1st
(b) 2nd
(c) 3rd

8. Deciles divide the data into _____ equal parts.
(a) six
(b) nine
(c) ten

9. The value below which 10% of the data falls is known as the _____ decile.
(a) 1st
(b) 5th
(c) 10th

10. Percentiles divide the data into _____ equal parts.
(a) 50
(b) 75
(c) 100

11. The median is also referred to as the _____ percentile.
(a) 50th
(b) 25th
(c) 75th

12. The value below which 1% of the data falls is called the _____ percentile.
(a) 10th
(b) 1st
(c) 5th

13. The mode is the value with the _____ frequency in a dataset.
(a) highest
(b) lowest
(c) equal

14. The graphical location of quartiles and percentiles is often marked on a(n) _____.
(a) scatter plot
(b) bar chart
(c) cumulative frequency curve

15. The mean is affected by extreme values, making it sensitive to _____.
(a) outliers
(b) medians
(c) modes

Example 1: Calculating Mean and Mode

Given the dataset: 12, 15, 18, 22, 18, 12, 10, 18

(a) Calculate the mean. (b) Identify the mode.

Solution: (a) Mean = (12 + 15 + 18 + 22 + 18 + 12 + 10 + 18) / 8 = 15.375 (b) The mode is 18, as it appears most frequently in the dataset.

Example 2: Finding the Median

Given the dataset: 8, 12, 15, 18, 22, 24

Find the median.

Solution: Since there are 6 values, the median is the average of the 3rd and 4th values: (15 + 18) / 2 = 16.5.

Example 3: Quartiles and Percentiles

Given the dataset: 7, 9, 10, 12, 14, 16, 18, 20, 22

(a) Calculate the 1st quartile. (b) Calculate the 75th percentile.

Solution: (a) Q1 = 1/4 * (n + 1) = 1/4 * (9) = 2.25 So, the 1st quartile is between the 2nd and 3rd values: (9 + 10) / 2 = 9.5 (b) P75 = 75/100 * (n + 1) = 0.75 * (9) = 6.75 The 75th percentile is between the 6th and 7th values: (16 + 18) / 2 = 17.

Example 4: Deciles and Mode

Given the dataset: 5, 7, 8, 10, 12, 15, 18, 20, 25, 28

(a) Calculate the 3rd decile. (b) Identify the mode.

Solution: (a) D3 = 3/10 * (n + 1) = 0.3 * (10) = 3 The 3rd decile is the 3rd value, which is 8. (b) The mode is not applicable in this dataset as no value appears more than once.

Example 5: Graphical Location of Mode and Median

Given the frequency distribution of a dataset:

(a) Determine the mode. (b) Identify the graphical location of the median.

Solution: (a) The mode is 15, as it has the highest frequency. (b) The median is the value where the cumulative frequency crosses half of the total frequency. In this case, it’s 15.

[mediator_tech]

Evaluation

1. The mean is also known as the _____.
(a) average
(b) mode
(c) median

2. The mode of a dataset is the value that appears _____ frequently.
(a) least
(b) most
(c) equally

3. The median divides a dataset into _____ halves.
(a) three
(b) unequal
(c) two equal

4. The mode can be graphically located at the _____ of a histogram.
(a) peak
(b) base
(c) center

5. Quartiles divide a dataset into _____ equal parts.
(a) three
(b) five
(c) four

6. The value below which 25% of the data falls is the _____ quartile.
(a) 3rd
(b) 1st
(c) 2nd

7. Deciles divide a dataset into _____ equal parts.
(a) nine
(b) ten
(c) eight

8. The value below which 10% of the data falls is the _____ decile.
(a) 10th
(b) 5th
(c) 1st

9. Percentiles divide a dataset into _____ equal parts.
(a) 75
(b) 50
(c) 100

10. The median is also referred to as the _____ percentile.
(a) 50th
(b) 25th
(c) 75th

11. The value below which 1% of the data falls is the _____ percentile.
(a) 1st
(b) 10th
(c) 5th

12. The graphical location of quartiles and percentiles is often marked on a(n) _____.
(a) scatter plot
(b) bar chart
(c) cumulative frequency curve

13. The mean is influenced by extreme values, making it sensitive to _____.
(a) medians
(b) modes
(c) outliers

14. The mode is the value with the _____ frequency in a dataset.
(a) highest
(b) lowest
(c) equal

15. The median is the middle value when the data is arranged in _____ order.
(a) random
(b) ascending
(c) descending

QUANTILES OF FRACTILES

TERMS USED IN FREQUENCY DISTRIBUTIONS