Table with Class Intervals (with Frequencies)
| County | Aggravated Assault Cases (𝑥) | Frequency (𝑓) | 𝑓 * 𝑥 |
|---|---|---|---|
| Allen | 10 | 1 | 10 |
| Belmont | 27 | 1 | 27 |
| Carroll | 2 | 1 | 2 |
| Clermont | 85 | 1 | 85 |
| Fairfield | 33 | 1 | 33 |
| Fulton | 7 | 1 | 7 |
| Geauga | 3 | 1 | 3 |
| Greene | 20 | 1 | 20 |
| Hamilton | 81 | 1 | 81 |
| Hocking | 16 | 1 | 16 |
| Jefferson | 9 | 1 | 9 |
| Lawrence | 39 | 1 | 39 |
| Licking | 34 | 1 | 34 |
| Lorain | 35 | 1 | 35 |
| Lucas | 67 | 1 | 67 |
| Madison | 6 | 1 | 6 |
| Mahoning | 0 | 1 | 0 |
| Medina | 7 | 1 | 7 |
| Miami | 26 | 1 | 26 |
| Montgomery | 157 | 1 | 157 |
| Perry | 4 | 1 | 4 |
| Pickaway | 17 | 1 | 17 |
| Portage | 304 | 1 | 304 |
| Richland | 20 | 1 | 20 |
| Stark | 59 | 1 | 59 |
| Summit | 44 | 1 | 44 |
| Union | 16 | 1 | 16 |
| Wood | 7 | 1 | 7 |
| 𝚺𝑓 = 28 | 𝚺𝑓𝑥 = 1135 |
Measures of Center
Mean (𝑥̄)
𝑥̄ = 𝚺𝑓𝑥 ÷ 𝚺𝑓
𝑥̄ = 1135 ÷ 28
𝑥̄ = 40.535714285714285
The average aggravated assault cases per county for all 28 counties is approximately 41 cases.
Median (𝑥̃)
𝚺𝑓 = 28
With the sample size being an even number, this means that the median will be between the 14th and 15th data values after sorting the data in ascending order.
Sorted data in ascending order:
0,
2,
3,
4,
6,
7,
7,
7,
9,
10,
16,
16,
17,
20,
20,
26,
27,
33,
34,
35,
39,
44,
59,
67,
81,
85,
157,
304
14th value (2019) = 20
15th value (2019) = 20
𝑥̃ = (𝑥14 + 𝑥15) ÷ 2
𝑥̃ = (20 + 20) ÷ 2
𝑥̃ = 40 ÷ 2
𝑥̃ = 20
The median number of aggravated assault cases in all 28 counties, after the data has been sorted in ascending order, is 20 cases of aggravated assault.
Mode (𝑥̂)
The mode is the data value that has the highest frequency/occurs the most often.
The number of cases of aggravated assault reported the most in 2019 was 7 cases of aggravated assault.
𝑥̂ = 7
Midrange (𝑥MR)
𝑥min = 0 cases
𝑥max = 304 cases
𝑥MR = (𝑥min + 𝑥max) ÷ 2
𝑥MR = (0 + 304) ÷ 2
𝑥MR = 152
Mahoning county had the lowest reported number of aggravated assault cases in 2019: 0 cases of aggravated assault.
Portage county had the highest reported number of aggravated assault cases in 2019: 304 cases of aggravated assault.
The midrange, or average of the minimum and maximum cases of aggravated assaults, is 152 cases of aggravated assault.
Aggravated Assault Cases in Ohio, 2019, sorted in Ascending Order
| 2019 Aggravated Assault Cases (𝑥) | Frequency (𝑓) | 𝑥 - 𝑥̄ | (𝑥 - 𝑥̄)2 | 𝑓(𝑥 - 𝑥̄)2 |
|---|---|---|---|---|
| 0 | 1 | -40.535714285714285 | 1643.1441326530612 | 1643.1441326530612 |
| 2 | 1 | -38.535714285714285 | 1485.001275510204 | 1485.001275510204 |
| 3 | 1 | -37.535714285714285 | 1408.9298469387754 | 1408.9298469387754 |
| 4 | 1 | -36.535714285714285 | 1334.8584183673468 | 1334.8584183673468 |
| 6 | 1 | -34.535714285714285 | 1192.7155612244896 | 1192.7155612244896 |
| 7 | 3 | -33.535714285714285 | 1124.6441326530612 | 3373.9323979591836 |
| 9 | 1 | -31.535714285714285 | 994.501275510204 | 994.501275510204 |
| 10 | 1 | -30.535714285714285 | 932.4298469387754 | 932.4298469387754 |
| 16 | 2 | -24.535714285714285 | 602.001275510204 | 1204.002551020408 |
| 17 | 1 | -23.535714285714285 | 553.9298469387754 | 553.9298469387754 |
| 20 | 2 | -20.535714285714285 | 421.71556122448976 | 843.4311224489795 |
| 26 | 1 | -14.535714285714285 | 211.28698979591834 | 211.28698979591834 |
| 27 | 1 | -13.535714285714285 | 183.21556122448976 | 183.21556122448976 |
| 33 | 1 | -7.535714285714285 | 56.78698979591835 | 56.78698979591835 |
| 34 | 1 | -6.535714285714285 | 42.71556122448978 | 42.71556122448978 |
| 35 | 1 | -5.535714285714285 | 30.644132653061213 | 30.644132653061213 |
| 39 | 1 | -1.5357142857142847 | 2.3584183673469354 | 2.3584183673469354 |
| 44 | 1 | 3.4642857142857153 | 12.001275510204088 | 12.001275510204088 |
| 59 | 1 | 18.464285714285715 | 340.92984693877554 | 340.92984693877554 |
| 67 | 1 | 26.464285714285715 | 700.358418367347 | 700.358418367347 |
| 81 | 1 | 40.464285714285715 | 1637.358418367347 | 1637.358418367347 |
| 85 | 1 | 44.464285714285715 | 1977.0727040816328 | 1977.0727040816328 |
| 157 | 1 | 116.46428571428572 | 13563.929846938778 | 13563.929846938778 |
| 304 | 1 | 263.4642857142857 | 69413.42984693879 | 69413.42984693879 |
| 𝚺𝑓 = 28 | 𝚺𝑓(𝑥 - 𝑥̄)2 = 103138.9642857143 |
Measures of Spread
Range
The range of a dataset is equal to the minimum value of the dataset subtracted from the maximum value of the dataset.
𝑥min = 0 cases
𝑥max = 304 cases
range = 𝑥max - 𝑥min
range = 304 - 0
range = 304
Mahoning county had the least amount of reported aggravated assault cases in 2019: 0 cases of aggravated assault.
Portage county had the highest amount of reported aggravated assault cases in 2019: 304 cases of aggravated assault.
The range of number of reported aggravated assault cases between Mahoning and Portage counties in 2019 is 304 cases of aggravated assault.
Variance (𝑠2)
Variance is a measurement that calculates the distance data values have from the mean; it is equal to the square of standard deviation.
𝚺𝑓𝑥 = 1135
𝚺𝑓 = 1135
𝑥̄ = 40.535714285714285 cases
𝑠2 = (𝚺𝑓(𝑥 - 𝑥̄)2) ÷ (𝚺𝑓 - 1)
𝑠2 = (103138.9642857143) ÷ (28 - 1)
𝑠2 = (103138.9642857143) ÷ (27)
𝑠2 = 3819.9616402116408
The variance of the dataset is approximately 3820. This means that there is medium to large spread between the data values and the mean.
Standard Deviation (𝑠)
Standard deviation is the measure of the distance the data values are from the mean. A larger number usually means more spread and variation between the values in the dataset.
𝑠 = √(𝑠2)
𝑠 = √(3819.9616402116408)
𝑠 = 61.80583823727044
The standard deviation is approximately 62. This means that there is medium to large spread between the values of the dataset from the mean.
Measures of Position
The measures of location are found after sorting the data in ascending order. There are 28 numbers in the dataset, which means there are 7 numbers in each quartile (divide the dataset into four parts).
First Quartile: 0 (Mahoning), 2 (Carroll), 3 (Geauga), 4 (Perry), 6 (Madison), 7 (Fulton), 7 (Medina)
Second Quartile: 7 (Wood), 9 (Jefferson), 10 (Allen), 16 (Hocking), 16 (Union), 17 (Pickaway), 20 (Greene)
Third Quartile: 20 (Richland), 26 (Miami), 27 (Belmont), 33 (Fairfield), 34 (Licking), 35 (Lorain), 39 (Lawrence)
Fourth Quartile: 44 (Summit), 59 (Stark), 67 (Lucas), 81 (Hamilton), 85 (Clermont), 157 (Montgomery), 304 (Portage)
The Lower Quartile (25th Percentile) is between 7 (Medina) and 7 (Wood)
Lower Quartile = (𝑥7 + 𝑥8) ÷ 2
Lower Quartile = (7 + 7) ÷ 2
Lower Quartile = 14 ÷ 2
Lower Quartile = 7
When the data for aggravated assault cases in metropolitan counties in Ohio, 2019, is sorted in ascending order, 7 cases of aggravated assault are in the lower quartile, 25 percent.
The Middle Quartile has already been calculated: it is the same as the median (𝑥̃ = 20).
Middle Quartile = 𝑥̃
Middle Quartile = 20
When the data for aggravated assault cases in metropolitan counties in Ohio, 2019, is sorted in ascending order, 20 cases of aggravated assault are in the middle quartile, 50 percent.
The Upper Quartile (75th Percentile) is between 39 (Lawrence) and 44 (Summit)
Upper Quartile = (𝑥21 + 𝑥22) ÷ 2
Upper Quartile = (39 + 44) ÷ 2
Upper Quartile = 83 ÷ 2
Upper Quartile = 41.5
When the data for aggravated assault cases in metropolitan counties in Ohio, 2019, is sorted in ascending order, approximately 41.5 cases are in the upper quartile, 75 percent.
Five-Number Summary
Q1 = 7
Q2 (median) = 20
Q3 = 41.5
Maximum = 304 (Portage)