Pergunta

Example: Find the median of the following distribution. Class & Frequency 40-44 & 7 45-49 & 10 50-54 & 22 55-59 & 15 60-64 & 12 65-69 & 6 70-74 & 3
Solução

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GiovanaMestre · Tutor por 5 anos
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### 54.16
Explicação
## Step1: Calculate the cumulative frequencies.
### We need to add a cumulative frequency column to the table. The cumulative frequency for each class is the sum of the frequencies of all classes up to and including that class.
## Step2: Determine the median class.
### The median is the middle value of the dataset. First, calculate the total frequency,
## Step3: Apply the median formula.
### The formula for the median of grouped data is:
###
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## Step4: Calculate the values for the formula.
### From the table, we can identify the values needed for the median formula.
## Step5: Substitute the values into the formula and calculate the median.
### Substitute the values obtained in the previous step into the median formula and perform the calculation.
**Detailed Solution:**
## Step1: Calculate the cumulative frequencies.
### The cumulative frequencies are calculated as follows:
### - 40-44: 7
### - 45-49: 7 + 10 = 17
### - 50-54: 17 + 22 = 39
### - 55-59: 39 + 15 = 54
### - 60-64: 54 + 12 = 66
### - 65-69: 66 + 6 = 72
### - 70-74: 72 + 3 = 75
## Step2: Determine the median class.
### The total frequency,
###
### The median class is the class that contains the 37.5-th value, which is the 50-54 class (since its cumulative frequency is 39, which is the first cumulative frequency greater than 37.5).
## Step3: Apply the median formula.
### The median formula is:
## Step4: Calculate the values for the formula.
### -
### -
### -
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### -
## Step5: Substitute the values into the formula and calculate the median.
###
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###
###
### We need to add a cumulative frequency column to the table. The cumulative frequency for each class is the sum of the frequencies of all classes up to and including that class.
## Step2: Determine the median class.
### The median is the middle value of the dataset. First, calculate the total frequency,
. Then, find the class that contains the
-th value. This class is the median class.
## Step3: Apply the median formula.
### The formula for the median of grouped data is:
###
, where:
### -
is the lower boundary of the median class.
### -
is the total frequency.
### -
is the cumulative frequency of the class preceding the median class.
### -
is the frequency of the median class.
### -
is the class width.
## Step4: Calculate the values for the formula.
### From the table, we can identify the values needed for the median formula.
## Step5: Substitute the values into the formula and calculate the median.
### Substitute the values obtained in the previous step into the median formula and perform the calculation.
**Detailed Solution:**
## Step1: Calculate the cumulative frequencies.
### The cumulative frequencies are calculated as follows:
### - 40-44: 7
### - 45-49: 7 + 10 = 17
### - 50-54: 17 + 22 = 39
### - 55-59: 39 + 15 = 54
### - 60-64: 54 + 12 = 66
### - 65-69: 66 + 6 = 72
### - 70-74: 72 + 3 = 75
## Step2: Determine the median class.
### The total frequency,
, is 75.
###
### The median class is the class that contains the 37.5-th value, which is the 50-54 class (since its cumulative frequency is 39, which is the first cumulative frequency greater than 37.5).
## Step3: Apply the median formula.
### The median formula is:
## Step4: Calculate the values for the formula.
### -
(lower boundary of the median class 50-54) = 49.5
### -
(total frequency) = 75
### -
(cumulative frequency of the class preceding the median class) = 17
### -
(frequency of the median class) = 22
### -
(class width) = 5
## Step5: Substitute the values into the formula and calculate the median.
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