Calculate Cumulative Frequency: To find the median of a frequency distribution, we first need to determine the cumulative frequency for each class interval.
Find Median Class Interval: We calculate the cumulative frequency by adding the frequency of each class interval to the sum of the frequencies of all previous intervals.
Calculate Median: The cumulative frequency for the first interval $(10-19)$ is $1$ .
Calculate Median: The cumulative frequency for the first interval $(10-19)$ is $1$ .The cumulative frequency for the second interval $(20-29)$ is $1$ (previous) $+ 6 = 7$ .
Calculate Median: The cumulative frequency for the first interval $(10-19)$ is $1$ .The cumulative frequency for the second interval $(20-29)$ is $1$ (previous) + $6 = 7$ .The cumulative frequency for the third interval $(30-39)$ is $7$ (previous) + $9 = 16$ .
Calculate Median: The cumulative frequency for the first interval $(10-19)$ is $1$ .The cumulative frequency for the second interval $(20-29)$ is $1$ (previous) + $6 = 7$ .The cumulative frequency for the third interval $(30-39)$ is $7$ (previous) + $9 = 16$ .The cumulative frequency for the fourth interval $(40-49)$ is $16$ (previous) + $1$ $0$ .
Calculate Median: The cumulative frequency for the first interval $(10-19)$ is $1$ .The cumulative frequency for the second interval $(20-29)$ is $1$ (previous) + $6 = 7$ .The cumulative frequency for the third interval $(30-39)$ is $7$ (previous) + $9 = 16$ .The cumulative frequency for the fourth interval $(40-49)$ is $16$ (previous) + $1$ $0$ .The cumulative frequency for the fifth interval $1$ $1$ is $1$ $2$ (previous) + $1$ $3$ .
Calculate Median: The cumulative frequency for the first interval $(10-19)$ is $1$ .The cumulative frequency for the second interval $(20-29)$ is $1$ (previous) + $6 = 7$ .The cumulative frequency for the third interval $(30-39)$ is $7$ (previous) + $9 = 16$ .The cumulative frequency for the fourth interval $(40-49)$ is $16$ (previous) + $1$ $0$ .The cumulative frequency for the fifth interval $1$ $1$ is $1$ $2$ (previous) + $1$ $3$ .The cumulative frequency for the sixth interval $1$ $4$ is $1$ $5$ (previous) + $1$ $6$ .
Calculate Median: The cumulative frequency for the first interval $(10-19)$ is $1$ .The cumulative frequency for the second interval $(20-29)$ is $1$ (previous) + $6 = 7$ .The cumulative frequency for the third interval $(30-39)$ is $7$ (previous) + $9 = 16$ .The cumulative frequency for the fourth interval $(40-49)$ is $16$ (previous) + $1$ $0$ .The cumulative frequency for the fifth interval $1$ $1$ is $1$ $2$ (previous) + $1$ $3$ .The cumulative frequency for the sixth interval $1$ $4$ is $1$ $5$ (previous) + $1$ $6$ .The cumulative frequency for the seventh interval $1$ $7$ is $1$ $8$ (previous) + $1$ $9$ .
Calculate Median: The cumulative frequency for the first interval $(10-19)$ is $1$ .The cumulative frequency for the second interval $(20-29)$ is $1$ (previous) + $6 = 7$ .The cumulative frequency for the third interval $(30-39)$ is $7$ (previous) + $9 = 16$ .The cumulative frequency for the fourth interval $(40-49)$ is $16$ (previous) + $1$ $0$ .The cumulative frequency for the fifth interval $1$ $1$ is $1$ $2$ (previous) + $1$ $3$ .The cumulative frequency for the sixth interval $1$ $4$ is $1$ $5$ (previous) + $1$ $6$ .The cumulative frequency for the seventh interval $1$ $7$ is $1$ $8$ (previous) + $1$ $9$ .The cumulative frequency for the eighth interval $(20-29)$ $0$ is $(20-29)$ $1$ (previous) + $(20-29)$ $2$ .
Calculate Median: The cumulative frequency for the first interval ( $10$ $-19$ ) is $1$ .The cumulative frequency for the second interval ( $20$ $-29$ ) is $1$ (previous) + $6 = 7$ .The cumulative frequency for the third interval ( $30$ $-39$ ) is $7$ (previous) + $9 = 16$ .The cumulative frequency for the fourth interval ( $40$ $-49$ ) is $16$ (previous) + $31 = 47$ .The cumulative frequency for the fifth interval ( $50$ $-59$ ) is $47$ (previous) + $42 = 89$ .The cumulative frequency for the sixth interval ( $60$ $-69$ ) is $89$ (previous) + $1$ $0$ .The cumulative frequency for the seventh interval ( $70$ $-79$ ) is $1$ $1$ (previous) + $1$ $2$ .The cumulative frequency for the eighth interval ( $80$ $-89$ ) is $1$ $3$ (previous) + $1$ $4$ .The cumulative frequency for the ninth interval ( $90$ $-99$ ) is $1$ $5$ (previous) + $1$ $6$ , which is the total number of observations.
Calculate Median: The cumulative frequency for the first interval $(10-19)$ is $1$ .The cumulative frequency for the second interval $(20-29)$ is $1$ (previous) + $6 = 7$ .The cumulative frequency for the third interval $(30-39)$ is $7$ (previous) + $9 = 16$ .The cumulative frequency for the fourth interval $(40-49)$ is $16$ (previous) + $1$ $0$ .The cumulative frequency for the fifth interval $1$ $1$ is $1$ $2$ (previous) + $1$ $3$ .The cumulative frequency for the sixth interval $1$ $4$ is $1$ $5$ (previous) + $1$ $6$ .The cumulative frequency for the seventh interval $1$ $7$ is $1$ $8$ (previous) + $1$ $9$ .The cumulative frequency for the eighth interval $(20-29)$ $0$ is $(20-29)$ $1$ (previous) + $(20-29)$ $2$ .The cumulative frequency for the ninth interval $(20-29)$ $3$ is $(20-29)$ $4$ (previous) + $(20-29)$ $5$ , which is the total number of observations.The median is the value that separates the higher half from the lower half of the data set. Since we have $(20-29)$ $6$ observations, the median will be the average of the $(20-29)$ $7$ th and $(20-29)$ $8$ th values.
Calculate Median: The cumulative frequency for the first interval $(10-19)$ is $1$ .The cumulative frequency for the second interval $(20-29)$ is $1$ (previous) $+ 6 = 7$ .The cumulative frequency for the third interval $(30-39)$ is $7$ (previous) $+ 9 = 16$ .The cumulative frequency for the fourth interval $(40-49)$ is $16$ (previous) $1$ $0$ .The cumulative frequency for the fifth interval $1$ $1$ is $1$ $2$ (previous) $1$ $3$ .The cumulative frequency for the sixth interval $1$ $4$ is $1$ $5$ (previous) $1$ $6$ .The cumulative frequency for the seventh interval $1$ $7$ is $1$ $8$ (previous) $1$ $9$ .The cumulative frequency for the eighth interval $(20-29)$ $0$ is $(20-29)$ $1$ (previous) $(20-29)$ $2$ .The cumulative frequency for the ninth interval $(20-29)$ $3$ is $(20-29)$ $4$ (previous) $(20-29)$ $5$ , which is the total number of observations.The median is the value that separates the higher half from the lower half of the data set. Since we have $(20-29)$ $6$ observations, the median will be the average of the $(20-29)$ $7$ th and $(20-29)$ $8$ th values.To find the class interval that contains the $(20-29)$ $7$ th and $(20-29)$ $8$ th values, we look for the cumulative frequency that is just greater than $(20-29)$ $7$ . This is the fifth interval $1$ $1$ with a cumulative frequency of $1$ $5$ .
Calculate Median: The cumulative frequency for the first interval $(10-19)$ is $1$ .The cumulative frequency for the second interval $(20-29)$ is $1$ (previous) + $6 = 7$ .The cumulative frequency for the third interval $(30-39)$ is $7$ (previous) + $9 = 16$ .The cumulative frequency for the fourth interval $(40-49)$ is $16$ (previous) + $1$ $0$ .The cumulative frequency for the fifth interval $1$ $1$ is $1$ $2$ (previous) + $1$ $3$ .The cumulative frequency for the sixth interval $1$ $4$ is $1$ $5$ (previous) + $1$ $6$ .The cumulative frequency for the seventh interval $1$ $7$ is $1$ $8$ (previous) + $1$ $9$ .The cumulative frequency for the eighth interval $(20-29)$ $0$ is $(20-29)$ $1$ (previous) + $(20-29)$ $2$ .The cumulative frequency for the ninth interval $(20-29)$ $3$ is $(20-29)$ $4$ (previous) + $(20-29)$ $5$ , which is the total number of observations.The median is the value that separates the higher half from the lower half of the data set. Since we have $(20-29)$ $6$ observations, the median will be the average of the $(20-29)$ $7$ th and $(20-29)$ $8$ th values.To find the class interval that contains the $(20-29)$ $7$ th and $(20-29)$ $8$ th values, we look for the cumulative frequency that is just greater than $(20-29)$ $7$ . This is the fifth interval $1$ $1$ with a cumulative frequency of $1$ $5$ .Now we need to calculate the median using the formula for grouped data: $\newline$ $\text{Median} = L + \left[\frac{N/2 - CF}{f}\right] * w$ $\newline$ where $1$ $4$ is the lower boundary of the median class, $1$ $5$ is the total number of observations, $1$ $6$ is the cumulative frequency of the class before the median class, $1$ $7$ is the frequency of the median class, and $1$ $8$ is the width of the median class.
Calculate Median: The cumulative frequency for the first interval $10-19$ is $1$ .The cumulative frequency for the second interval $20-29$ is $1$ (previous) + $6 = 7$ .The cumulative frequency for the third interval $30-39$ is $7$ (previous) + $9 = 16$ .The cumulative frequency for the fourth interval $40-49$ is $16$ (previous) + $1$ $0$ .The cumulative frequency for the fifth interval $1$ $1$ is $1$ $2$ (previous) + $1$ $3$ .The cumulative frequency for the sixth interval $1$ $4$ is $1$ $5$ (previous) + $1$ $6$ .The cumulative frequency for the seventh interval $1$ $7$ is $1$ $8$ (previous) + $1$ $9$ .The cumulative frequency for the eighth interval $20-29$ $0$ is $20-29$ $1$ (previous) + $20-29$ $2$ .The cumulative frequency for the ninth interval $20-29$ $3$ is $20-29$ $4$ (previous) + $20-29$ $5$ , which is the total number of observations.The median is the value that separates the higher half from the lower half of the data set. Since we have $20-29$ $6$ observations, the median will be the average of the $20-29$ $7$ th and $20-29$ $8$ th values.To find the class interval that contains the $20-29$ $7$ th and $20-29$ $8$ th values, we look for the cumulative frequency that is just greater than $20-29$ $7$ . This is the fifth interval $1$ $1$ with a cumulative frequency of $1$ $5$ .Now we need to calculate the median using the formula for grouped data: $\newline$ Median = $1$ $4$ $\newline$ where $1$ $5$ is the lower boundary of the median class, $1$ $6$ is the total number of observations, $1$ $7$ is the cumulative frequency of the class before the median class, $1$ $8$ is the frequency of the median class, and $1$ $9$ is the width of the median class.The lower boundary ( $1$ $5$ ) of the median class $1$ $1$ is $6 = 7$ $2$ . $\newline$ The total number of observations ( $1$ $6$ ) is $20-29$ $6$ . $\newline$ The cumulative frequency ( $1$ $7$ ) before the median class is $1$ $2$ . $\newline$ The frequency ( $1$ $8$ ) of the median class is $6 = 7$ $8$ . $\newline$ The width ( $1$ $9$ ) of the class intervals is $30-39$ $0$ (since each interval spans $10$ units, e.g., $10-19$ , $20-29$ , etc.).
Calculate Median: The cumulative frequency for the first interval $10-19$ is $1$ .The cumulative frequency for the second interval $20-29$ is $1$ (previous) + $6 = 7$ .The cumulative frequency for the third interval $30-39$ is $7$ (previous) + $9 = 16$ .The cumulative frequency for the fourth interval $40-49$ is $16$ (previous) + $1$ $0$ .The cumulative frequency for the fifth interval $1$ $1$ is $1$ $2$ (previous) + $1$ $3$ .The cumulative frequency for the sixth interval $1$ $4$ is $1$ $5$ (previous) + $1$ $6$ .The cumulative frequency for the seventh interval $1$ $7$ is $1$ $8$ (previous) + $1$ $9$ .The cumulative frequency for the eighth interval $20-29$ $0$ is $20-29$ $1$ (previous) + $20-29$ $2$ .The cumulative frequency for the ninth interval $20-29$ $3$ is $20-29$ $4$ (previous) + $20-29$ $5$ , which is the total number of observations.The median is the value that separates the higher half from the lower half of the data set. Since we have $20-29$ $6$ observations, the median will be the average of the $20-29$ $7$ th and $20-29$ $8$ th values.To find the class interval that contains the $20-29$ $7$ th and $20-29$ $8$ th values, we look for the cumulative frequency that is just greater than $20-29$ $7$ . This is the fifth interval $1$ $1$ with a cumulative frequency of $1$ $5$ .Now we need to calculate the median using the formula for grouped data: $\newline$ Median = $1$ $4$ $\newline$ where $1$ $5$ is the lower boundary of the median class, $1$ $6$ is the total number of observations, $1$ $7$ is the cumulative frequency of the class before the median class, $1$ $8$ is the frequency of the median class, and $1$ $9$ is the width of the median class.The lower boundary ( $1$ $5$ ) of the median class $1$ $1$ is $6 = 7$ $2$ . $\newline$ The total number of observations ( $1$ $6$ ) is $20-29$ $6$ . $\newline$ The cumulative frequency ( $1$ $7$ ) before the median class is $1$ $2$ . $\newline$ The frequency ( $1$ $8$ ) of the median class is $6 = 7$ $8$ . $\newline$ The width ( $1$ $9$ ) of the class intervals is $30-39$ $0$ (since each interval spans $30-39$ $0$ units, e.g., $10-19$ , $20-29$ , etc.).Substitute the values into the formula: $\newline$ Median = $30-39$ $4$ $\newline$ Median = $30-39$ $5$ $\newline$ Median = $30-39$ $6$ $\newline$ Median = $30-39$ $7$ $\newline$ Median = $30-39$ $8$ $\newline$ Median = $30-39$ $9$ $\newline$ Median = $7$ $0$
Calculate Median: The cumulative frequency for the first interval ( $10$ $-19$ ) is $1$ .The cumulative frequency for the second interval ( $20$ $-29$ ) is $1$ (previous) + $6 = 7$ .The cumulative frequency for the third interval ( $30$ $-39$ ) is $7$ (previous) + $9 = 16$ .The cumulative frequency for the fourth interval ( $40$ $-49$ ) is $16$ (previous) + $31 = 47$ .The cumulative frequency for the fifth interval ( $50$ $-59$ ) is $47$ (previous) + $42 = 89$ .The cumulative frequency for the sixth interval ( $60$ $-69$ ) is $89$ (previous) + $1$ $0$ .The cumulative frequency for the seventh interval ( $70$ $-79$ ) is $1$ $1$ (previous) + $1$ $2$ .The cumulative frequency for the eighth interval ( $80$ $-89$ ) is $1$ $3$ (previous) + $1$ $4$ .The cumulative frequency for the ninth interval ( $90$ $-99$ ) is $1$ $5$ (previous) + $1$ $6$ , which is the total number of observations.The median is the value that separates the higher half from the lower half of the data set. Since we have $1$ $7$ observations, the median will be the average of the $1$ $8$ th and $1$ $9$ th values.To find the class interval that contains the $1$ $8$ th and $1$ $9$ th values, we look for the cumulative frequency that is just greater than $1$ $8$ . This is the fifth interval ( $50$ $-59$ ) with a cumulative frequency of $89$ .Now we need to calculate the median using the formula for grouped data: $\newline$ Median = $6 = 7$ $4$ $\newline$ where $6 = 7$ $5$ is the lower boundary of the median class, $6 = 7$ $6$ is the total number of observations, $6 = 7$ $7$ is the cumulative frequency of the class before the median class, $6 = 7$ $8$ is the frequency of the median class, and $6 = 7$ $9$ is the width of the median class.The lower boundary ( $6 = 7$ $5$ ) of the median class ( $50$ $-59$ ) is $7$ $1$ . $\newline$ The total number of observations ( $6 = 7$ $6$ ) is $1$ $7$ . $\newline$ The cumulative frequency ( $6 = 7$ $7$ ) before the median class is $47$ . $\newline$ The frequency ( $6 = 7$ $8$ ) of the median class is $7$ $7$ . $\newline$ The width ( $6 = 7$ $9$ ) of the class intervals is $7$ $9$ (since each interval spans $7$ $9$ units, e.g., $10$ $-19$ , $20$ $-29$ , etc.).Substitute the values into the formula: $\newline$ Median = $9 = 16$ $1$ $\newline$ Median = $9 = 16$ $2$ $\newline$ Median = $9 = 16$ $3$ $\newline$ Median = $9 = 16$ $4$ $\newline$ Median = $9 = 16$ $5$ $\newline$ Median = $9 = 16$ $6$ $\newline$ Median = $9 = 16$ $7$ The median of the given frequency distribution is $9 = 16$ $7$ .