# Which measure of central tendency can be determined graphically?

In statistical data, the descriptive summary of a data set is known as the central tendency. It is considered to be an indicator of the center of the data distribution. To summarise, it represents the single value of the entire distribution or a dataset. It describes the data precisely.

However, the central tendency of the data distribution does not provide information regarding individual data from the dataset. There are various measures of the central tendency of the data.

**Measures of Central Tendency **

Since we described, the central tendency of the data as an indicator of the values, describing the dataset by computing the central position of the data. The central tendency of the data can be measured by the following quantities:

**Mean:**Mean is considered to be the sum of all observations divided by the total number of observations.**Median:**Median is defined as the middle or central value in an ordered set.**Mode:**Mode is defined as the most frequently occurring value in a data set.

The measure of central tendency that can be determined graphically is median

**Merits or Uses of Median:**

- Median is considered to be rigidly defined like Mean
- Median remains unaffected even in case the value of the extreme item is much different from other values. For instance, the Median in the case of the given numbers, 4, 7, 12, 18, 19 is 12. In case, we add two integral values equivalent to 450 and 10000, the new median obtained is 18.
- Can be used for Quantities.
- Can be located graphically
- In the case of the open-end intervals, calculating the median is the perfect option.
- Easy to understand.
- Easy to calculate.
- Used for statistical devices such as Mean Deviation and skewness.
- Can be calculated by inspection.

For instance, let us assume that we have some of the terms missing and the middle four terms are given to be 4, 5, 7, 15. The median can then be calculated as :

* | * | 4 | 5 | 7 | 15 | * | * | * |

Here out of nine terms middle term is;

Therefore, 7 is the median of the given data.

**Demerits or Limitations of Median:**

- In case the extremes’ value is significantly large, the median cannot be used as the representative of the series.
- It is affected much more by fluctuations of sampling than A.M.
- In the case of a continuous series, the median has to be interpolated.
- Median is not suitable to be used for algebraic treatment.
- The value of median is only be computed for continuous series if the frequencies are uniformly spread over the whole class interval in which the median is spread.
- In case, the number of series given is even, an estimate can be done. This is the A.M. of two middle terms, which then becomes the median.

**Graphical Method**

Marks | Conversion into exclusive series | No. of students | Cumulative Frequency |

(x) |
| (f) | (C.M) |

410 – 419 | 409.5 – 419.5 | 14 | 14 |

420 – 429 | 419.5 – 429.5 | 20 | 34 |

430 – 439 | 429.5 – 439.5 | 42 | 76 |

440 – 449 | 439.5 – 449.5 | 54 | 130 |

450 – 459 | 449.5 – 459.5 | 45 | 175 |

460 – 469 | 459.5 – 469.5 | 18 | 193 |

470 – 479 | 469.5 – 479.5 | 7 | 200 |

The median value of a given data distribution can be determined through the graphic visualization of data in the form of ogives.

This is possible in two ways:

- Less than ogive
- More than ogive

**Steps involved in calculating median using less than Ogive approach:**

Marks | Cumulative Frequency (C.M) |

Less than 419.5 | 14 |

Less than 429.5 | 34 |

Less than 439.5 | 76 |

Less than 449.5 | 130 |

Less than 459.5 | 175 |

Less than 469.5 | 193 |

Less than 479.5 | 200 |

- The series is converted into a more than cumulative frequency distribution.
- Let us assume N to be the total number of students.
- The cumulative frequency of the last interval is also considered to be N.
- Compute the value of item(student) and mark it on the y-axis. In this case the item (students) is = 100
^{th}student. - A perpendicular is drawn from 100 to the right to intersect the Ogive curve at the marked point A in the graph.
- Similarly, a perpendicular is drawn on the x-axis at the cut point of the ogive curve.
- The point at which the sts is touched will be the corresponding median value.

**Steps involved in calculating median using more than Ogive approach. **

Let us consider the following data distribution:

Marks | Cumulative Frequency (C.M) |

More than 409.5 | 200 |

More than 419.5 | 186 |

More than 429.5 | 166 |

More than 439.5 | 124 |

More than 449.5 | 70 |

More than 459.5 | 25 |

More than 469.5 | 7 |

More than 479.5 | 0 |

- The series is converted into a more than cumulative frequency distribution.
- Let us assume N to be the total number of students.
- The cumulative frequency of the last interval is also considered to be N.
- Compute the value of (student) and mark it on the y-axis.In this case the item (student) is = 100
^{th}student - A perpendicular is drawn from 100 to the right to intersect the Ogive curve at the marked point A in the graph.
- Similarly, a perpendicular is drawn on the x-axis at the cut point of the ogive curve.
- The point at which the sts is touched will be the corresponding median value.

### Sample Problems

**Question 1. Find the median of the data graphically?**

Class Interval | Frequency |

0-10 | 06 |

10-20 | 12 |

20-30 | 22 |

30-40 | 16 |

40-50 | 12 |

**Solution:**

For more than cumulative frequency

Class Interval Cumulative Frequency More than 0 68 More than 10 62 More than 20 50 More than 30 28 More than 40 12 For less than cumulative frequency

Class Interval Cumulative Frequency Less than 10 6 Less than 20 18 Less than 30 40 Less than 40 56 Less than 50 68 Thus for the median

Median is where both the more than and less than curves meets

i.e, 27.5

Therefore the median is 27.5.

**Question 2. Find the median of the ungrouped data**

**Assume the data: **

**55, 78, 52, 59, 86, 77, 62**

**Solution:**

First arrange the data in ascending order

52, 55, 59, 62, 77, 78, 86

As we can see that here total number of observations are 7

Thus,

Median = = 4

Therefore,

Here the Median is 4

^{th}observationThe 4

^{th}observation is 59

**Question 3. Find the median of the ungrouped data**

**Assume the data;**

**15, 18, 9, 5, 7, 23 **

**Solution:**

First arrange the data in ascending order

5, 7, 9, 15, 18, 23

As we can see that here are 6 observations

So the formula for median when the observations are in even

If number of observations is even,

Thus use this formula:

Median =

Thus according to the formula

Median = 12

**Question 4. Find the median of the grouped data**

**The distribution is**

Classes | 0-20 | 20-40 | 40-60 | 60-80 | 80-100 |

Frequency | 4 | 24 | 44 | 16 | 12 |

**Solution:**

We need to calculate the cumulative frequencies to find the median.

Classes Number of students Cumulative frequency 0-20 4 4 20-40 24 4 + 24 = 28 40-60 44 28 + 44 = 72 60-80 16 72 + 16 = 88 80-100 12 88 + 12 = 100 Here,

N = 100

Median Class = (40 – 60)l = 40, f = 44, c = 28, h = 20

By apply grouped data median formula

Therefore,

Median = 50

**Question 5. Compute the median of the following data distribution : **

**10, 42, 78, 18, 36, 5, 19, 45, 69, 75, 26, 17, 20, 29, 31**

**Solution:**

Arranging the data in ascending order

5, 10, 17, 18, 19, 20, 26, 29, 31, 36, 42, 45, 69, 75, 78

As we can see that here total number of observations are 15

Thus,

Median = = 8

Therefore,

Here the Median is 8

^{th}observationThe 8

^{th}observation is 29.

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