Statistics – How to Find Histogram Mean

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Im trying to find the mean of the histogram below:

I found that the sum of the frequencies is $110$, so I then get that the mean should be $$\frac{19 \cdot 20 + 20 \cdot 30 + 10(21+22+23+25+26+27)}{110} = 22$$ however the mean should be between $22$ and $23$ according to the solution sheet. Any ideas on what Im doing wrong?

Best Answer

Use the midpoint of each bar, not the left of each bar. In other words, use $19.5$ rather than $19$; use $20.5$ rather than $20$, etc. Note that this makes sense because the histogram says that there are $20$ observations between $19$ and $20$, so we assume these observations are evenly distributed throughout the class $19 < x < 20$, and so the average of these $20$ observations is $19.5.$

Also, take note that we assume these observations are evenly distributed throughout the class, which is an assumption, i.e. it may be false. But this is what me must do with grouped data: we must make assumptions in order to calculate any statistics (e.g. the mean or interquartile range) from the data.

Related Solutions

[Math] How to find mean and median from histogram

You can get both the mean and the median from the histogram. The way to calculate the mean is that illustrated in the video and already shown in one of the comments. For each histogram bar, we start by multiplying the central x-value to the corresponding bar height. Each of these products corresponds to the sum of all values falling within each bar. Summing all products gives us the total sum of all values, and dividing it by the number of observations yields the mean.

On the other hand, to calculate the median from a histogram you have to apply the following classical formula:

$$\displaystyle L_m + \left [ \frac { \frac{N}{2} - F_{m-1} }{f_m} \right ] \cdot c$$

where $L_m$ is the lower limit of the median bar, $N$ is the total number of observations, $F_{m-1}$ is the cumulative frequency of the bar preceding the median bar (i.e. the total number of observations in all bars below the median bar), $f$ is the frequency of the median bar, and $c$ is the median bar width. This formula substantially arises from a linear interpolation, which assumes that data are uniformly distributed within the median class. To understand this formula, it can be noted that the fraction $\displaystyle\frac {N/2 - F_{m-1}}{f_m}$ is the proportion of observations in the median bar that are below the median. Under the assumption that observations are uniformly distributed within the median bar, multiplying this proportion by the median bar width $c$ yields the fraction of median bar width corresponding to the position of the median. Adding this result to $L_m$ finally provides the median.

[Math] How to find median from a histogram

Add up all the frequencies to find the total number of whatever it is ($n$). Find $\dfrac{n+1}{2}$, and that's the element you need to find the value of.

Now you just need to iterate over the histogram. Keep a running total of frequencies. When your total passes $\dfrac{n+1}{2}$, the last value you added the frequency for is the median.

In python, if you have the histogram as a dictionary (in your example, {5: 0, 10: 36, 15: 54, 20: 69, 25: 82, 30: 55, 35: 43, 40: 25, 45: 22, 50: 17, 55: 0}),

def median(histogram)
    total = 0
    median_index = (sum(histogram.values()) + 1) / 2
    for value in sorted(histogram.keys()):
        total += histogram[value]
        if total > median_index:
            return value

Best Answer

Related Solutions

[Math] How to find mean and median from histogram

[Math] How to find median from a histogram

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