13.4.2. Example 1

A survey regarding the heights (in cm) of 51 girls of Class X of a school was conducted and the following data was obtained: Height (in cm) Number of girls Less than 140 4 Less than 145 11 Less than 150 29 Less than 155 40 Less than 160 46 Less than 165 51

In this chunk, we are looking at the cumulative data about the heights of 51 girls. The table shows how many girls have heights less than certain values. To find the median height, we need to first determine which class interval contains the median value. Since there are 51 girls, the median will be the value at the 25.5th position (which means we need to find the values at the 25th and 26th position). By examining the cumulative frequencies, we see that the cumulative frequency for the height less than 150 cm is 29, which includes both the 25th and 26th positions, indicating that this is our median class.

Think of it like a race where you have a group of 51 runners. If you want to find out the runner who finishes in the middle position, you might look for the 25th and 26th finishers to see who is in the middle of the pack. In terms of the heights, we are essentially trying to see what height reflects that middle position among those girls.

Now, to calculate the median height, we need to find the class intervals and their corresponding frequencies. The given distribution being of the less than type, 140, 145, 150, . . ., 165 give the upper limits of the corresponding class intervals. So, the classes should be below 140, 140 - 145, 145 - 150, . . ., 160 - 165. Observe that from the given distribution, we find that there are 4 girls with height less than 140, i.e., the frequency of class interval below 140 is 4. Now, there are 11 girls with heights less than 145 and 4 girls with height less than 140. Therefore, the number of girls with height in the interval 140 - 145 is 11 – 4 = 7.

In this step, we redefine the class intervals based on the data given and calculate the frequencies for each interval. For example, if 4 girls are shorter than 140 cm, and 11 girls are shorter than 145 cm, this means there are 7 girls whose heights fall between 140 and 145 cm. We repeat this process for each subsequent interval to complete our frequency distribution table. This distribution enables us to see how many girls fit into each height range, which is critical when calculating the median.

Imagine you host a height competition. After measuring everyone, you note how many participants fall below certain heights. Just like finding out how many players are in specific height ranges helps you understand your competition better, defining your height ranges in this problem helps in understanding how the heights of girls are distributed.

n = 51. Now n = 51. So, = 25.5. This observation lies in the class 145 - 150. Then, l (the lower limit) = 145, cf (the cumulative frequency of the class preceding 145 - 150) = 11, f (the frequency of the median class 145 - 150) = 18, h (the class size) = 5. Using the formula, Median = l + (n/2 - cf) × h.

Now that we've located the median class (145 - 150), we apply the median formula. In this case, the lower limit (l) of this class is 145, the cumulative frequency (cf) of the class before it is 11, the frequency (f) of the median class is 18, and the class width (h) is 5. By substituting these values into our median formula: Median = l + ((n/2 - cf) / f) x h, we can solve for the median height of the girls.

Just like using a recipe to bake cookies, you need specific measurements to achieve the perfect cookie. Here, the recipe is the median formula, and the measurements (lower limit, cumulative frequency, frequency, and class size) help you calculate the average height accurately, giving an overall representation of the class.

Substituting the values: Median = 145 + (25.5 - 11) × 5 / 18 = 145 + 14.5 / 18 = 145 + 0.8055 ≈ 149.03. So, the median height of the girls is 149.03 cm.

Now, we take the values we determined and calculate the median height step-by-step. By plugging the numbers into the formula we set up earlier, we find that the median height of these girls is 149.03 cm. This value means that about half the girls are shorter than this height, and half are taller. The calculation reflects a central value among the group.

Imagine a school of fish swimming together. The median height, 149.03 cm, serves as the 'middle' fish, so to speak, where half of the fish are smaller and half are larger. Thus, it gives you a middle point in the school’s overall height.

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