Calculate the quartile deviation for the given grouped data.

To calculate the quartile deviation for the given grouped data, we first need to find the first quartile (Q_1) and the third quartile (Q_3). The formula for the k^th quartile (Q_k) for grouped data is: Q_k = L + ((kn)/(4) - cfbf)w Where: L = Lower boundary of the k^th quartile class n = Total frequency ( f) cfb = Cumulative frequency of the class preceding the k^th quartile class f = Frequency of the k^th quartile class w = Class width First, let's prepare the data by finding the class boundaries and cumulative frequencies. The given class intervals are discrete, so we convert them to continuous class boundaries by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit. | Class Interval | Class Boundaries | Frequency (f) | Cumulative Frequency (cf) | | :------------- | :--------------- | :-------------- | :------------------------ | | 20-21 | 19.5-21.5 | 4 | 4 | | 22-23 | 21.5-23.5 | 12 | 4 + 12 = 16 | | 24-25 | 23.5-25.5 | 24 | 16 + 24 = 40 | | 26-27 | 25.5-27.5 | 12 | 40 + 12 = 52 | | 28-29 | 27.5-29.5 | 4 | 52 + 4 = 56 | From the table: Total frequency n = f = 56. Class width w = 21.5 - 19.5 = 2. Step 1: Calculate the First Quartile (Q_1). The position of Q_1 is (1n)/(4) = (1 × 56)/(4) = 14^th item. The 14^th item falls in the class with a cumulative frequency of 16, which is the 22-23 class (class boundaries 21.5-23.5). For this class: L = 21.5 cfb = 4 (cumulative frequency of the preceding class) f = 12 (frequency of the Q_1 class) w = 2 Substitute these values into the formula: Q_1 = 21.5 + ((14 - 4)/(12))2 Q_1 = 21.5 + ((10)/(12))2 Q_1 = 21.5 + (20)/(12) Q_1 = 21.5 + 1.666... Q_1 ≈ 23.17 Step 2: Calculate the Third Quartile (Q_3). The position of Q_3 is (3n)/(4) = (3 × 56)/(4) = 3 × 14 = 42^nd item. The 42^nd item falls in the class with a cumulative frequency of 52, which is the 26-27 class (class boundaries 25.5-27.5). For this class: L = 25.5 cfb = 40 (cumulative frequency of the preceding class) f = 12 (frequency of the Q_3 class) w = 2 Substitute these values into the formula: Q_3 = 25.5 + ((42 - 40)/(12))2 Q_3 = 25.5 + ((2)/(12))2 Q_3 = 25.5 + (4)/(12) Q_3 = 25.5 + 0.333... Q_3 ≈ 25.83 Step 3: Calculate the Interquartile Range (IQR). The Interquartile Range is the difference between the third and first quartiles: IQR = Q_3 - Q_1 IQR = 25.83 - 23.17 IQR = 2.66 Step 4: Calculate the Quartile Deviation (QD). The Quartile Deviation is half of the Interquartile Range: QD = (IQR)/(2) QD = (2.66)/(2) QD = 1.33 The quartile deviation is 1.33. Send me the next one 📸