The following table gives the number of days in June and...

The following table gives the number of days in June and July of recent years in which the temperature reached 90 degrees or higher in New York’s Central Park. Source: The New York Times and Accuweather.com. a. Prepare a frequency distribution with a column for intervals and frequencies. Use six intervals, starting with 0–4. b. Sketch a histogram and a frequency polygon, using the intervals in part a. c. Find the mean for the original data. d. Find the mean using the grouped data from part a. e. Explain why your answers to parts c and d are different. f. Find the median and the mode for the original data.

The following table gives the number of days in June and July of recent years in which the temperature reached 90 degrees or higher in New York’s Central Park. Source: The New York Times and Accuweather.com.
a. Prepare a frequency distribution with a column for intervals and frequencies. Use six intervals, starting with 0–4.
b. Sketch a histogram and a frequency polygon, using the intervals in part a.
c. Find the mean for the original data.
d. Find the mean using the grouped data from part a.
e. Explain why your answers to parts c and d are different.
f. Find the median and the mode for the original data.

Key Concepts

Grouped vs Ungrouped Data

Ungrouped data consists of individual data points, allowing more precise calculations of statistical measures. Grouped data, on the other hand, is compiled into intervals to simplify large datasets, but this aggregation can introduce approximation errors. Understanding the differences between these methods is crucial because the process of grouping can affect the accuracy of measures like the mean.

Mean Calculations

The mean is a measure of central tendency that provides the average value of a dataset. It can be computed directly from the raw, ungrouped data, or estimated from grouped data. When calculated from grouped data, the mean is approximated using the midpoints of the intervals, which may lead to slight differences compared to the mean of the original data.

Histograms

A histogram is a type of bar graph that represents the frequency distribution of a dataset. Each bar in the histogram corresponds to an interval from the frequency distribution, and its height reflects the frequency of data points within that interval. Histograms provide a visual summary of data variation and are useful for identifying patterns such as skewness and modality.

Frequency Distribution

A frequency distribution is a way to organize and summarize data by dividing the range of values into intervals (or bins) and counting the number of observations that fall into each interval. This method allows for a clear overview of how data points are distributed over the range and is essential for further statistical analysis and graphical representation.

Frequency Polygons

A frequency polygon is a graphical tool that displays the frequencies of intervals with a line graph by connecting the midpoints of the tops of the bars in a histogram. This method is helpful in comparing different distributions and understanding trends in the data over continuous intervals.

Transcript

00:01 This question shows us a table that describes how many days reached 90 degrees in new york's central park.

00:09 First, it wants us to make a frequency table.

00:14 Group these by bins of length 5, starting with 0 to 4.

00:19 So we'll have 0 to 4, 5 to 9, 10 to 14, and i want 6 of them.

00:26 So we'll go up to 15 to 19, 20 to 24, and finally 25 to 29.

00:36 Now, the way i'm going to do this is i'm going to go through, and for each one, i'll just make a little mark by the bin that the number falls in.

00:46 So we start 11 -8 -11 -38.

00:50 So to do that, i will make a mark where 11 goes here, 10 to 14, then a mark where 8 goes, 11 -8, 11.

00:58 Then 3 and then 8 and i'll keep doing that and i've actually already done it and i'll find that the frequencies we have are 9 in the first bin 9 in the second 14 1 5 and 1 so these are our frequencies now wants me to make a histogram and a frequency polygon so i'm gonna draw my axes here and the way i'm gonna say i'm gonna say set up my bins is i'm going to have them left endpoint inclusive.

01:33 So this is going to be zero.

01:34 And if this is four, the bar that's here will include zero but not, excuse me, i'll have this to be five.

01:42 So the bar here will include zero, but not five.

01:45 And that's what we want, because the next bar will include five.

01:50 So if we do that, i will have my bins set up like this, zero, five, ten, fifteen, twenty, twenty, five, five.

02:03 And 30.

02:06 And i'll choose to scale my x -axis up to, let's just go by two is up to seven.

02:12 So two, four, six, eight, ten, twelve, fourteen, i meant fourteen, i'm in fourteen, not seven.

02:23 All right, and so now we can start drawing this out.

02:26 So in our first bin, our zero to four we had nine.

02:29 So that'll be two, four, six, eight, and up to nine right there.

02:34 And our second base, we also had 9, so that'll be right here.

02:39 In our third we had 14.

02:41 It's all the way up to the top.

02:43 And 14 there.

02:45 In our fourth we only had 1.

02:48 Then a spike up to 5 and then back down to 1.

02:54 From here we can draw our frequency polygon.

02:56 This is just connecting the midpoints of the tops of each bar.

03:02 So it'll look something like this.

03:04 It's hand drawn, so it's not going to be perfect.

03:07 But we'll have a curve that looks like this.

03:10 This will give us a, oops, whoa, it'll give us a shape that sort of describes the shape of how these numbers are distributed.

03:22 Now, the next thing it wants us to do is find the mean.

03:29 What we do to find the mean, we know is we sum up all of our data, all of our x's into five, but add the number...

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